The Theory of Splines and Their Applications (Mathematics in Science and Engineering 38)

The Theory of Splines and Their Applications

M A T H E MAT I C S I N SCIENCE AND ENGINEERING ~~

A SERIES OF M O N O G R A P H S A N D T E X T B O O K S

Edited by Richard Bellman University of Southern California 1.

2. 3. 4.

5. 6. 7. 8. 9.

i0. 11. 12. 13. 14. 15. 16. 17.

18. 19. 20. 21. 22.

TRACY Y. THOMAS. Concepts from Tensor Analysis and Differential Geometry. Second Edition. 1965 TRACY Y.THOMAS. Plastic Flow and Fracture in Solids. 1961 RUTHERFORD ARIS. The Optimal Design of Chemical Reactors: A Study in Dynamic Programming. 1961 JOSEPH LASALLEand SOLOMON LEFSCHETZ. Stability by Liapunov’s Direct Method with Applications. 1961 GEORGE LEITMANN (ed.) . Optimization Techniques : With Applications to Aerospace Systems. 1962 RICHARDBELLMANand KENNETHL. COOKE.Differential-Difference Equations. 1963 Mathematical Theories of Traffic Flow. 1963 FRANKA. HAIGHT. F. V. ATKINSON. Discrete and Continuous Boundary Problems. 1964 Non-Linear Wave Propagation: With AppliA. JEFFREY and T. TANIUTI. cations to Physics and Magnetohydrodynamics. 1964 JULIUS T. Tou. Optimum Design of Digital Control Systems. 1963 HARLEY FLANDERS. Differential Forms: With Applications to the Physical Sciences. 1963 SANFORD M. ROBERTS. Dynamic Programming in Chemical Engineering and Process Control. 1964 SOLOMON LEPSCHETZ. Stability of Nonlinear Control Systems. 1965 DIMITRISN. CHORAFAS. Systems and Simulation. 1965 A. A. PERVOZVANSKII. Random Processes in Nonlinear Control Systems. 1965 MARSHALL C. PEASE,111. Methods of Matrix Algebra. 1965 V. E. BENES.Mathematical Theory of Connecting Networks and Telephone Traffic. 1965 WILLIAM F. AMES.Nonlinear Partial Differential Equations in Engineering. 1965 J. A C Z ~ LLectures . on Functional Equations and Their Applications. 1966 R. E. MURPHY.Adaptive Processes in Economic Systems. 1965 S. E. DREYFUS.Dynamic Programming and the Calculus of Variations. 1965 A. A. FEL’DBAUM. Optimal Control Systems. 1965

MATHEMATICS 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38.

IN S C I E N C E A N D ENGINEERING

A. HALANAY. Differential Equations: Stability, Oscillations, Time Lags.

1966 M. NAMIKOGIUZTORELI. Time-Lag Control Systems. 1966 DAVIDSWORDER. Optimal Adaptive Control Systems. 1966 MILTONASH. Optimal Shutdown Control of Nuclear Reactors. 1966 DIMITRIS N. CHORAFAS. Control System Functions and Programming Approaches. (In Two Volumes.) 1966 N. P. ERUGIN.Linear Systems of Ordinary Differential Equations. 1966 SOLOMON MARCUS.Algebraic Linguistics ; Analytical Models. 1967 A. M. LIAPUNOV. Stability of Motion. 1966 GEORGELEITMANN (ed.). Topics in Optimization. 1967 MASANAO AOKI. Optimization of Stochastic Systems. 1967 HAROLD J. KUSHNER.Stochastic Stability and Control. 1967 MINORUURABE.Nonlinear Autonomous Oscillations. 1967 F. CALOGERO. Variable Phase Approach to Potential Scattering. 1967 A. K A U F M A N Graphs, N. Dynamic Programming, and Finite Games. 1967 A. K A U F M A Nand N R. CRUON. Dynamic Programming: Sequential Scientific Management. 1967 J. H. AHLBERG, E. N. NILSON,and J. L. WALSH.The Theory of Splines and Their Applications. 1967

In prepavation Y . SAWARAGI, Y . S U N A ~ A Rand A , T. NAKAMIZO. Statistical Decision Theory in Adaptive Control Systems A. KAUFMANN and R. FAURE.Introduction to Operations Research RICHARD BELLMAN. Introduction to the Mathematical Theory of Control Processes ( I n Three Volumes.) E. STANLEY LEE. Quasilinearization and Invariant Bedding WILLARDMILLER,JR. Lie Theory and Special Functions F. SHAMPINE, and PAULE. WALTMAN. Nonlinear PAULB. BAILEY,LAWRENCE Two Point Boundary Value Problems

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The Theory of Splines and Their Applications J . H . AHLBERG UNITED AIRCRAFT RESEARCH LABORATORIES EASTHARTFORD, CONNECTICUT

E . N . NILSON PRATT & WHITNEY AIRCRAFT COMPANY EASTHARTFORD, CONNECTICUT

J . L. WALSH UNIVERSITY OF MARYLAND COLLEGE PARK,MARYLAND

1967

ACADEMIC PRESS

New York and London

COPYRIGHT 0 1967,

BY

ACADEMIC PRESSINC.

ALL RIGHTS RESERVED. NO PART OF THIS BOOK MAY BE REPRODUCED I N ANY FORM, BY PHOTOSTAT, MICROFILM, OR ANY OTHER MEANS, WITHOUT WRITTEN PERMISSION FROM THE PUBLISHERS.

ACADEMIC PRESS INC. 1 1 1 Fifth Avenue, New York, New York 10003

United Kingdom Edition published by ACADEMIC PRESS INC. (LONDON) LTD. Berkeley Square House, London W.l

LIBRARY OF CONGRESS CATALOG CARDNUMBER: 66-30115

PRINTED I N THE UNITED STATES OF AMERICA

PREFACE

Spline functions constitute a relatively new subject in analysis. During the past decade both the theory of splines and experience with their use in numerical analysis have undergone a considerable degree of development. Discoveries of new and significant results are of frequent occurrence. It is useful at this juncture, nevertheless, to make some serious effort to organize and present material already developed up to this time. Much of this has become standardized. On the other hand, there are several areas where the theory is not yet complete. This book contains much of the material published since 1956 together with a considerable amount of the authors’ own research not previously presented; it also reflects a considerable amount of practical experience with splines on the part of the authors. I n the interests of holding the present volume to a reasonable size, certain areas related to splines have been omitted. T h u s the work of Schoenberg and his associates o n the use of splines in the smoothing of equidistant data has not been included, nor is there any treatment of the theory of splines of complex argument. We hope, nevertheless, that the material presented will provide the reader with the necessary background for both theoretical and applied work in what promises to be a very active and extensive area. I n Chapter I there is a brief description of what is meant by a spline; this is followed by a survey of the development of spline theory since 1946 when Schoenberg first introduced the concept of a mathematical spline. We develop in Chapters I1 and IV, respectively, the theory of cubic splines and polynomial splines of higher degree from an algebraic point of view; the methods employed depend heavily on the equations used to define the spline. I n particular, these chapters contain much of the material basic for applications. I n Chapters I11 and V we reconsider cubic and polynomial splines of higher degree from a different point of view which reveals more clearly their deeper structure. Although the resulting theorems are not so sharp as their counterparts in Chapters 11 and IV, they are more easily carried over to new settings. This is done vii

...

Vlll

PREFACE

in Chapters VI, VII, and VIII, in which we consider in turn generalized splines, doubly cubic splines, and two-dimensional generalized splines. We wish to express our deep gratitude to all those who have contributed to making this book a reality. Specifically, we wish to thank the United Aircraft Research Laboratories, the Pratt & Whitney Division of the United Aircraft Corporation, Harvard University, and the University of Maryland, whose support has made possible much of our research in spline theory.

May, 1967

J. H. AHLBERG E. N. NILSON J. L. WALSH

CONTENTS

PREFACE.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii

Introduction

Chapter I

1.1. What Is a Spline ? . . . . . . . . . . . . . . 1.2. Recent Developments in the Theory of Splines

Chapter I1 2.1. 2.2. 2.3. 2.4. 2.5. 2.6. 2.7. 2.8. 2.9.

Chapter I11

...... . . . . . . .

The Cubic Spline Introduction . . . . . . . . . . . . . . . . . . . . . .

Existence. Uniqueness. and Best Approximation . Convergence . . . . . . . . . . . . . . . . Equal Intervals . . . . . . . . . . . . . . . Approximate Differentiation and Integration . . . Curve Fitting . . . . . . . . . . . . . . . Approximate Solution of Differential Equations . Approximate Solution of Integral Equations . . Additional Existence and Convergence Theorems

. . . . . . . . . . . . . . . . . . . . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

Intrinsic Properties of Cubic Splines The Minimum Norm Property . . . . . . . . . . . . . . The Best Approximation Property . . . . . . . . . . . . . The Fundamental Identity . . . . . . . . . . . . . . . . The First Integral Relation . . . . . . . . . . . . . . . . Uniqueness . . . . . . . . . . . . . . . . . . . . . . Existence . . . . . . . . . . . . . . . . . . . . . . . . General Equations . . . . . . . . . . . . . . . . . . . Convergence of Lower-Order Derivatives . . . . . . . . . The Second Integral Relation . . . . . . . . . . . . . . . .

3.1. 3.2. 3.3. 3.4. 3.5. 3.6. 3.7. 3.8. 3.9. 3.10. Raising the Order of Convergence . . . . 3.1 1. Convergence of Higher-Order Derivatives 3.12. Limits on the Order of Convergence . . . 3.13. Hilbert Space Interpretation . . . . . . 3.14. Convergence in Norm . . . . . . . . . 3.15. Canonical Mesh Bases and Their Properties 3.1 6. Remainder Formulas . . . . . . . . . . 3.17. Transformations Defined by a Mesh . . . 3.18. A Connection with Space Technology . . ix

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . .

. . . . . . . . . . . . . . . . . .

1 2

9 16 19 34 42 50 52 57 61

75 77 78 79 82 84 84 87 89 91 93 95 97 98 101 103 105 107

CONTENTS

X

Chapter I V 4.1. 4.2. 4.3. 4.4. 4.5. 4.6.

The Polynomial Spline Definition and Working Equations . . . . . . . . . . . . Equal Intervals . . . . . . . . . . . . . . . . . . . . . Existence . . . . . . . . . . . . . . . . . . . . . . . . Convergence . . . . . . . . . . . . . . . . . . . . . . Quintic Splines of Deficiency 2, 3 . . . . . . . . . . . . . Convergence of Periodic Splines on Uniform Meshes . . . . .

109 124 132 135 143 148

Intrinsic Properties of Polynomial Splines of Odd Degree

Chapter V 5.1. 5.2. 5.3. 5.4. 5.5. 5.6. 5.7. 5.8. 5.9. 5.10. 5.1 1. 5.12. 5.13. 5.14. 5.15. 5.16. 5.17. 5.18.

Chapter VI 6.1. 6.2. 6.3. 6.4. 6.5. 6.6. 6.7. 6.8. 6.9. 6.10. 6.1 1. 6.12. 6.13. 6.14. 6.15. 6.16.

Introduction . . . . . . . . . . . . . . . . . . . . . . The Fundamental Identity . . . . . . . . . . . . . . . The First Integral Relation . . . . . . . . . . . . . . . The Minimum Norm Property . . . . . . . . . . . . . The Best Approximation Property . . . . . . . . . . . . Uniqueness . . . . . . . . . . . . . . . . . . . . . . Defining Equations . . . . . . . . . . . . . . . . . . Existence . . . . . . . . . . . . . . . . . . . . . . . . Convergence of Lower-Order Derivatives . . . . . . . . The Second Integral Relation . . . . . . . . . . . . . . Raising the Order of Convergence . . . . . . . . . . . . Convergence of Higher-Order Derivatives . . . . . . . . Limits on the Order of Convergence . . . . . . . . . . . Hilbert Space Interpretation . . . . . . . . . . . . . . Convergence in Norm . . . . . . . . . . . . . . . . . Canonical Mesh Bases and Their Properties . . . . . . . Kernels and Integral Representations . . . . . . . . . . Representation and Approximation of Linear Functionals . .

153 154 155 156 157 159 . 160 165 . 166 . 168 . 170 . 172 . 174 . 174 . 176 . 179 . 182 . 185

. . . .

Generalized Splines Introduction . . . . . . . . . . . . . . . . . . . . . . The Fundamental Identity . . . . . . . . . . . . . . . . The First Integral Relation . . . . . . . . . . . . . . . . The Minimum Norm Property . . . . . . . . . . . . . . Uniqueness . . . . . . . . . . . . . . . . . . . . . . Defining Equations . . . . . . . . . . . . . . . . . . . Existence . . . . . . . . . . . . . . . . . . . . . . . . Best Approximation . . . . . . . . . . . . . . . . . . Convergence of Lower-Order Derivatives . . . . . . . . . The Second Integral Relation . . . . . . . . . . . . . . . Raising the Order of Convergence . . . . . . . . . . . . . Convergence of Higher-Order Derivatives . . . . . . . . . Limits on the Order of Convergence . . . . . . . . . . . . Hilbert Space Interpretation . . . . . . . . . . . . . . . Convergence in Norm . . . . . . . . . . . . . . . . . . Canonical Mesh Bases . . . . . . . . . . . . . . . . .

191 192 193 195 196 197 199 200 201 204 206 208 211 213 216 219

xi

CONTENTS

6.17. Kernels and Integral Representations . . . . . . . . . . . 6.18. Representation and Approximation of Linear Functionals . . .

Chapter VII 7.1. 7.2. 7.3. 7.4. 7.5. 7.6. 7.7. 7.8. 7.9. 7.10. 7.1 1. 7.12. 7.13. 7.14. 7.15. 7.16,

220 221

The Doubly Cubic Spline Introduction . . . . . . . . . . . . . . . . . . Partial Splines . . . . . . . . . . . . . . . . . Relation of Partial Splines to Doubly Cubic Splines The Fundamental Identity . . . . . . . . . . . The First Integral Relation . . . . . . . . . . . The Minimum Norm Property . . . . . . . . . Uniqueness and Existence . . . . . . . . . . . Best Approximation . . . . . . . . . . . . . Cardinal Splines . . . . . . . . . . . . . . . . . Convergence Properties . . . . . . . . . . . . The Second Integral Relation . . . . . . . . . . T h e Direct Product of Hilbert Spaces . . . . . . The Method of Cardinal Splines . . . . . . . . . Irregular Regions . . . . . . . . . . . . . . . . Surface Representation . . . . . . . . . . . . The Surfaces of Coons . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . .

.

.

. .

235 237 238 240 242 242 243 244 245 247 248 249 251 254 258 262

Chapter VIII Generalized Splines in Two Dimensions 8.1. 8.2. 8.3. 8.4. 8.5. 8.6. 8.7. 8.8. 8.9.

. . . . . . . . 265 . . . . . . . . 266 . . . . . . . . . 267 . . . . . . . . 269 . . . . . . . . . 270 . . . . . . . . 271 . . . . . . . . 272 . . . . . . . . 274 . . . . . . . . . 275

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

277

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

281

Bibliography

INDEX.

Introduction . . . . . . . . . . . . . . Basic Definition . . . . . . . . . . . . The Fundamental Identity . . . . . . . Types of Splines . . . . . . . . . . . . The First Integral Relation . . . . . . . Uniqueness . . . . . . . . . . . . . . Existence . . . . . . . . . . . . . . . . Convergence . . . . . . . . . . . . . . Hilbert Space Theory . . . . . . . . .

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CHAPTER I

Introduction

1.1. What Is a Spline?

It seems appropriate to begin a book on spline theory by defining a spline in its simplest and most widely used form, and also to indicate the motivation leading to this definition. For many years, long, thin strips of wood or some other material have been used much like French curves by draftsmen to fair in a smooth curve between specified points. These strips or splines are anchored in place by attaching lead weights called “ducks” at points along the spline. By varying the points where the ducks are attached to the spline itself and the position of both the spline and the duck relative to the drafting surface, the spline can be made to pass through the specified points provided a sufficient number of ducks are used. If we regard the draftsman’s spline as a thin beam, then the BernoulliEuler law M ( x ) = EI[l/R(x)]

is satisfied. Here M ( x ) is the bending moment, E is Young’s modulus,

I is the geometric moment of inertia, and R ( x ) is the radius of curvature of the elastica, i.e., the curve assumed by the deformed axis of the beam. For small deflections, R(x) is &placed by l/y”(x),where y ( x ) denotes the elastica. T hu s we have y”(x) = (l/EI)M(x).

Since the ducks act effectively as simple supports, the variation of M ( x ) between duck positions is linear. T h e mathematical spline is the result of replacing the draftsman’s spline by its elastica and then approximating the latter by a piecewise cubic (normally a different cubic between each pair of adjacent ducks) with certain discontinuities of derivatives permitted at the junction points (the ducks) where two cubics join. 1

2

I.

INTRODUCTION

I n its simple form, the mathematical spline is continuous and has both a continuous first derivative and a continuous second derivative. Normally, however, there is a jump discontinuity in its third derivative at the junction points. This corresponds to the draftsman’s spline having continuous curvature with jumps occurring in the rate of change of curvature at the ducks. For many important applications, this mathematical model of the draftsman’s spline is highly realistic. I n practice, the draftsman does not place the ducks at the specified points through which his splin’e must pass. Moreover, there is not usually a one-to-one correspondence between the specified points and the ducks. On the other hand, when the mathematical analog is used, it is common practice to interpolate to the specified points at the junction points and to keep the number of specified points and junction points (including the endpoints) the same. I n the next section, we outline the recent history of the mathematical spline approximation. From this history, some of the properties of the mathematical spline become evident. Also, a considerable extension of the concept of a spline from that approximating the draftsman’s tool is apparent.

1.2. Recent Developments in the Theory of Splines T h e spline approximation in its present form first appeared in a paper by Schoenberg [ 19461.* As indicated in Section 1 . 1 , there is a very close relationship between spline theory and beam theory. Sokolnikoff [1956, pp. 1-41 provides a brief but very readable account of the development of beam theory. From the latter, one might anticipate some of the recent developments in the theory of splines, particularly the minimum curvature property. As suggested in Schoenberg’s paper [ 19461, approximations employed in actuarial work also frequently involve concepts that relate them closely to the spline. After 1946, Schoenberg, together with some of his students, continued these investigations of splines and monosplines. I n particular, Schoenberg and Whitney [1949; 19531 first obtained criteria for the existence of certain splines of interpolation. For the case of splines of even order with interpolation at the junction points, a simpler approach to the question of existence due to Ahlberg, Nilson, and Walsh [1964; 19651 is now possible; it makes use of a basic integral relation obtained for cubic

*

Data in square brackets refer to items in the Bibliography.

1.2.

RECENT DEVELOPMENTS I N THE THEORY OF SPLINES

3

splines of interpolation to a function f ( x )on a mesh A by Holladay [1957] which asserts

I” a

If”(x)

l2

dx =

a

1 S i ( f ;x) la dx

+ J’ If”(x) a

-

S i ( f ;x) l a dx.

Here S,( f ; x) denotes the spline of interpolation to f f x ) on A . I n this book, we refer to this integral relation as the first integral relation. T h e establishment of the first integral relation for certain cubic splines of interpolation was Holladay’s proof of the following theorem.

Theorem (Holladay). Let A : a = x,, < x1 < < xN = b and a set of real numbers {yi} (i = 0, I , ..., N ) be given. Then of all functions f ( x ) having a continuous second derivative on [a, b] and such that f ( x i ) = yi (i = 0, I , ..., N ) , the spline function S,( f ; x) with junction points at the xi and with Si(f ; a) = Si(f;6 ) = 0 minimizes the integral -.a

(1.2.1)

Much of the present-day theory of splines began with this theorem and its proof. Since the integral (1.2.1) is often a good approximation to the integral of the square of the curvature for a curve y = f ( x ) , the content of Holladay’s theorem is often called the minimum curvature property. Its close relation to the minimization of potential energy of a deflected beam is apparent. * I n this book, we consider a number of generalizations of the simple cubic spline. I n these generalizations, there are analogs of Holladay’s theorem; but, since there is no relation to curvature in these new settings, we use the name minimum norm property instead. This is meaningful, since in each case there is an associated Hilbert space, for now denoted by 2,in which (1.2.1) or its counterpart is the square of the norm of f ( x ) . It was not until 1964 that the Hilbert space aspect of spline theory evolved. At that time, the authors (Ahlberg, Nilson, and Walsh [abs. 1964; 19641) introduced some orthonormal bases for the space 2 which consisted entirely of splines or, somewhat more precisely, of equivalence classes of splines. I n terms of any orthonormal basis for 2, a functionffx) in % has, of course, for any positive integer N , a best approximation by linear combinations of the first N basis elements.

* T h e potential energy of a statically deflected beam is equal to the work done on the beam to produce the deflections; this in turn is proportional to the integral of the square of the curvature of the elastica of the beam (cf. Sokolnikoff [1956, p. 21).

4

I.

INTRODUCTION

If II * llx denotes the norm of 2, Ui(x) (i = 1, 2, ...) denotes the basis elements, and

U(4

N

=

2a,Ui(x),

(1.2.2)

i=l

then l l f - U is minimized when ai is the coefficient of U,(x) in the expansion of f ( x ) in terms of the complete basis. It is desirable to have another characterization of this best approximation, particularly if the alternative characterization facilitates its determination. Such a characterization is available. I n 1962, the authors (Walsh, Ahlberg, and Nilson [1962]) obtained the result: given a mesh A : a = x, < x1 < < x N = b, then of all simple periodic cubic splines on A the spline that interpolates to a periodic function f ( x ) at the mesh points furnishes the best approximation in the preceding sense. Since then, a number of extensions of this result have been obtained: Ahlberg, Nilson, and Walsh [abs. 1963; abs. 1964a,b; 1964; 19651, deBoor [1963], Schoenberg [1964c], Greville [1964a], and deBoor and Lynch [abs. 1964; 19661. I n Chapters 111, V, and VI of this book, we develop the HiIbert space theory of splines for cubic splines, polynomial splines of odd degree, and generalized splines, respectively. We define 2 as a function space of classes of functions; we then show that 2 is a Hilbert space with respect to an appropriate choice of norm. T h e symbol 2, however, is replaced by other notation. T h e convergence of the spline approximations Sy)(f ;x ) to the as the mesh norm jl A /I = maxi approximated functionsf - xj I approaches zero has also come under close scrutiny. T h e first results were obtained by the authors (Walsh, Ahlberg, and Nilson [1962]) for cubic splines and utilized the first integral relation. Under the assumption that f ( x ) is in C2[a,b], it was shown for splines of interpolation to f ( x ) at the mesh points that S y ) ( f ;x ) converges uniformly to f ' e ) ( ~ for ) 01 = 0, 1. A more detailed analysis was made by Ahlberg and Nilson [abs. 1961; 1962; 19631. I n particular, it was shown that, if f ( x ) is in C2[a,b], then Si(f;x) converges uniformly to f " ( x )provided the mesh spacing approaches uniformity as I/ A 11 approaches zero. This mesh restriction was later removed by Sharma and Meir [abs. 1964; 19661. For f ( x ) in C4[a,b], Birkhoff and deBoor [abs. 1964; 19641 have shown that (1.2.3) .( = 0, 1, ..., 4) i p ( x ) - sy(f;X) 1 < K IA ~114-= provided the ratio Rd = maxill A / I / xi - xiPl I is bounded. On the other hand, for weaker restrictions on f ( x ) such as f ( x ) is in C[a, b] or

1.2.

RECENT DEVELOPMENTS I N THE THEORY OF SPLINES

5

f(x) is in Cl [ a,b], appropriate convergence properties have been obtained by Ahlberg, Nilson, and Walsh [abs. 19661. I n addition, the convergence of polynomial splines of odd degree has been investigated by Ahlberg, 'Nilson, and Walsh Cabs. 1963; 19651, Schoenberg [1964], and Ziegler [abs. 19651; the convergence of multidimensional splines by Ahlberg, Nilson, and Walsh [abs. 1964a; 1964; 1965al; and the convergence of generalized splines by the same authors [abs. 1964b; 1964; 1965a1. Many of these convergence results depend on the fine structure of the linear system of equations defining the spline. I n Chapters I1 and IV, we develop spline theory from this point of view. On the other hand, a number of convergence results can be established without appeal to the defining equations. I n particular, for polynomial splines of degree 2n - 1 this can be done with respect to the convergence of derivatives through order n - 1. Moreover, with the aid of the integral relation

which was established by Ahlberg, Nilson, and Walsh [1965a] under a variety of conditions, convergence of derivatives through order 2n - 2 can be established. This was shown for generalized splines, polynomial splines of odd degree being a special case of the latter. We refer to the integral relation (1.2.4) as the second integral relation; on a suitable function space, it is a manifestation of the Riesz theorem concerning the representation of linear functionals. I n Chapters 111, V, and VI, this approach is developed. T h e basic result is -

S y ) ( f ;x) 1

< K 11 d 1 2n-a-1

(a = 0,

1 ,..., 2n - 1)

(1.2.5)

provided RA is bounded. For cubic splines, this result is weaker than (1.2.3). Whether 2n - 01 - 1 can be replaced in general by 2n - 01 is an open question. T h e theory of splines has been extended in a number of directions. Of considerable importance is the extension to several dimensions. A start was made by Birkhoff and Garabedian [1960], but the first truly successful extension was made by deBoor [abs. 1961; 19621, who demonstrated (deBoor [1962]) both the existence and uniqueness of certain bicubic splines* of interpolation. Later Ahlberg, Nilson, and Walsh [abs. 1964a; 1965bI extended the first integral relation to splines in several dimensions. As a result, existence, uniqueness, the minimum

* We employ henceforth the terminology j f t , s) is Q doubte cubic rather thanf(t, s) is a bicubic to imply thatf(t, s) is a cubic in t for each s and a cubic in s for each t . A doubZy cubic s p h e is a double cubic in each subrectangle defined by a two-dimensional mesh.

6

I.

INTRODUCTION

norm property, and the best approximation property were obtained for a variety of multidimensional splines. Questions of convergence were reduced to similar questions in one dimension, for which answers were known. I n Chapters VII and VIII, we consider multidimensional splines. Another direction of generalization has been the replacement of the operator DZn associated with a polynomial spline of degree 2n - 1 (here D z dldx) by the operator L*L, where

and L* is the formal adjoint of L. I n each mesh interval, a spline S ( x ) now satisfies the equation L*LS = 0 rather than the equation D2nS = 0. Splines defined in this manner are called generalized splines. T h e first step in this direction was taken by Schoenberg [I 964~1,who considered L< trigonometric splines." T h e complete generalization followed: Greville [1964]; Ahlberg, Nilson, and Walsh [abs. 196413; 1964; 1965al; deBoor and Lynch [abs. 19641. A more abstract approach to spline theory has been made by Atteia [1965] and his colleagues at Grenoble. T h e operator L = D(D -- u) has been considered by Schweikert [1966], who termed the resulting splines "splines in tension." When u is properly chosen, these splines have some advantages over cubic splines, as well as some disadvantages; in particular, they tend to suppress the occurrence of inflexion points not indicated by the data but concentrate the curvature near the junction points. Generalized splines are the subject matter of Chapters VI and VIII. T h e approximation of a linear functional9 by a second linear functional 9 such that the remainder W = 9 - 2 annihilates polynomials of degree n - 1 has been given considerable attention by Sard [1963]. Under reasonable restrictions, (1.2.7)

the kernel X ( 9 ;t ) is called a Peano kernel. Sard [1963] has sought to determine 9 such that jlX(B?;t ) 2 dt

is minimized, and for a variety of functionals 9he has so determined 9. In 1964, Schoenberg succeeded in showing that, for 9 of the form

9 f = @of(Jco) + .J(Jcl>

+ ..-+

1.2.

RECENT DEVELOPMENTS I N THE THEORY OF SPLINES

7

and with mild restrictions on 2,the optimum 9 results when L?j = 9 S d ( f ;x), whereS,(f; x) is the simple(see be1ow)polynomial spline (of degree 2n - 1) interpolating tof(x) onfl: a = xo < x1 < < xN = 6 . This result has been generalized (Ahlberg, Nilson, and Walsh [abs. 19651; Ahlberg and Nilson [1966]) to &’ of the form

p < n - 2 and certain a,,j = 0 ab initio. For generalized splines, (1.2.7) becomes

with

9f=

b

X ( 9 ;t)Lf(t)dt,

a

and the kernel X ( & ;t ) depends on L (cf. also deBoor and Lynch [1966]). We consider these matters in Chapters 111, V, VI, and VIII. T h e generalization of Schoenberg’s results for approximating functionals required the introduction of splines of a somewhat different character. T h e following terminology facilitates a partial indication of the nature of these differences. A spline of order 2n is simple when there is at most a jump discontinuity in the (2n - 1)th derivative at a mesh point. I n most instances, the splines under consideration are simple splines. When jumps in derivatives of order greater than 2n - k - 1 are permitted at an interior mesh point xi , the spline is said to be of dejiciency k at xi. If the spline is of deficiency k at all interior mesh points, it is said to be of deficiency k. We impose, however, the restriction 0 k n. I n this terminology, a solution of L*Lf = 0 in [a, b] has deficiency zero, and a simple spline has deficiency one. T h e requirement that certain aij in (1.2.8) vanish ab initio often imposes even more complicated and irregular continuity requirements on the splines employed. Such splines are called heterogenous splines and are considered in detail in Chapters VI and VIII. They were introduced by Ahlberg, Nilson, and Walsh [abs. 19651 for studying the approximation of linear functionals. T h e work of Golomb and Weinberger on “optimal’’ approximation of linear functionals (cf. Golomb and Weinberger [1959]) is very closely related to spline theory. I n many instances the functions zi(x) entering

< <

8

I.

INTRODUCTION

into their approximations are splines of interpolation. Nevertheless this is not pointed out in the above reference ; the development proceeds along different lines and the functions n ( x ) appear only as solutions of a variational problem. Holladay’s theorem which makes the direct connection to spline theory apparent is not mentioned. More recently, however, Secrest [ 19661 has recognized the relationship.

CHAPTER I1

The Cubic Spline

2.1. Introduction

T h e natural starting point for a study of spline functions is the cubic spline. Its close relation with the draftsman’s spline that results from the thin-beam approximation leads to many of its important properties and motivates much of its application to problems in numerical analysis. T h e spline proves to be an effective tool in the elementary processes of interpolation and approximate integration. An outstanding characteristic, however, is its effectiveness in numerical differentiation. T o a considerable extent, this is a consequence of the strong convergence properties that it possesses. On the other hand, the best approximation and minimum norm properties developed in Chapter111 are also important contributors to this effectiveness. I n this introductory section, the basic working equations of the spline are given, together with the procedures necessary for the common applications of the spline. These are followed in Sections 2.2 and 2.3 by a presentation of the existence and convergence properties of greatest interest. I n Sections 2.4 and 2.5, equal-interval splines are discussed and special formulas presented for numerical differentiation and integration for this case. I n Section 2.6, the application to the solution of linear differential equations is introduced. T h e chapter is concluded with a derivation of convergence and existence properties that require more incisive methods of analysis. x b, and I n the mathematical spline, we consider an interval a subdivide it by a mesh of points corresponding to the locations of the ducks: A: a = xo < X I < < X N = b.

< <

An associated set of ordinates is prescribed: y:

YO ,y1 ,.*.,Y N .

We seek a function S,( Y ;x), which we shall denote by S,(X) or SA , y (x ) when there is no ambiguity (and by S A , ywhen the argument x is 9

10

11.

THE CUBIC SPLINE

suppressed) which is continuous together with its first and second derivatives on [a, 61, coincides with a cubic in each subinterval xj-l x xi (j= 1, 2,..., N ) , and satisfies S,( Y ;xi) = yi ( j = 0, 1,..., N ) . T h e function S,( Y ; x), or S,(x), is said to be a spline with respect to the mesh A , or a spline on A , interpolating to the values yi at the mesh locations. T h e spline is said to be periodic of period ( b - a ) if the condition

< <

SY)(a+)

=

SY’(b-)

(P = 0 , 1 , 2 )

is satisfied. It is traditional to designate by M i the moment” Si(xi) ( j = 0, 1,..., N ) , even though this is not the true moment of the beam in the usual sense of the word. Th u s we have on [ x ~ - xi] ~ , from the linearity of the second derivative the equation ( I

(2.1.1)

If we integrate twice and evaluate the constants where hj = xi of integration, we obtain the equations S,(X)

=

x)3 6hj

(Xj -

+Mj

(x - x j - 1 ) 3 6hj

6 (2.1.2)

From (2.1.3) we have, for the one-sided limits of the derivative, the expressions

and S,(X) are conI n virtue of (2.1.1) and (2.1.2), the functions S~(X) ) by means of (2.1.4) tinuous on [a, b]. T h e continuity of s > (at~xjyields the condition

2.1.

11

INTRODUCTION

For the periodic spline, Eqs. (2.1.5) (j= 1, 2,..., N - 1) give N - 1 simultaneous equations in the quantities M , , M , ,..., M , . We require in this case that (2.1.5) be valid for j = N as well. Here y N = y o , M N = Mo , and we prescribe yN+l = y l , M,,, = M , , h,,, = h, . For the nonperiodic spline, two additionaI conditions must be 1 quantities specified, the “end conditions,” to determine the N Mo , M , ,..., M , . Specifying the slope of the spline at a and b gives the analog of the doubly cantilevered beam. For S>(a) = y b and Si(b) = yX we obtain from (2.1.4) the relations

+

Setting Mo = 0 and M N = 0 corresponds to placing simple supports at the ends. T h e condition Mo-AMl=O,

I>h>O

is equivalent to placing a simple support at x-, = (xo - Ax,)/( 1 - A) and requiring that the entire curve over x-, x x1 be the arc of a cubic. A common choice of A is i. We are generally concerned with end conditions, which, for convenience, we write in the form

< <

2Mo

+ AoM, = do,

pNMN-1

+~

M= N dN

(2.1.6)

*

We introduce the notation

( j = 1, 2,..*,N

hj

-

T h e continuity requirement (2.1.5) then becomes

I

For the nonperiodic spline, the defining equations (2.1.6) and (2.1.7) are now written as 2

A0

2

P2

0 0 0

... ... ...

0 0

0 0

0

0

0

0

0 -

... ...

...

2 PN-1

k

z

0

2

AN-

PN

2

,

,

(2.1.8)

11.

12

THE CUBIC SPLINE

where dj (j= 1, 2, ..., N - 1) represents the right-hand member of (2.1.7). For the periodic spline, the defining equations are 2

A,

0

...

0

0

0

."

0

0

."

0

0

3

PN

+

with M , = MNand AN = hl/(hN hl), p N = 1 - A N . For many applications, it is more convenient to work with the slopes mj = SL(xj) rather than with the moments M i .Here we find that on xj] we have the equations S,(x)

= mi-l

Si(x) = mi-l

(Xj

- x)"x

(xj

-

hi2

-

xj-1)

-

mi (x

- Xj_l)Z(Xj

hi2

- x)

+

+

x)(2x,4 xj - 3x) - mi (x - Xj4)(2Xj xj-1 - 3x) hi2 hi2

Moreover, the second derivative takes the form

(2.1.12) so that the limiting values from the two sides of xi are

T h e continuity condition is here imposed ( j = 1, 2, ..., N - 1). There results the requirement

on S:(x)

at xi

(2.1.14)

2.1.

13

INTRODUCTION

or, in more convenient form, the equation

IF1

T h e equation system for the nonperiodic spline is, therefore,

(2.1.16)

mN-2

mN-1 mN

-1

+

where the general end conditions 2m, + porn, = co , hNmN-l 2mN = c N are employed. T h e quantity cj ( j= 1, 2,..., N - 1) represents the right-hand member of (2.1.15). For the periodic case, the equations are

.

(2.1.17)

T h e end conditions are now classified in terms of the M j as follows: at

ti)

2%

+M, 2M0

(ii) (iii) 2 M ,

=

6 Y1 -Yo h, ( hl

= 2y;,

+ XoMl = do , at

(i)

x = a:

MN-1

x

=

(2.1.18)

b:

+ 2 M N = h6N ~

(yh

2MN = 2 y i ,

(ii)

+ 2M,

(iii) p N M N - ,

= dN ,

-

11.

14

THE CUBIC SPLINE

and, in terms of the m i , x

at

(ii) (iii)

+ m, = 3 Y1 hl-Yo 2mo + porn, = co , 2m0

x

at

(4 (ii) (iii)

= a:

=

hl

- YYO" >

b:

(2.1.1 9)

2mN = 2 y i ,

+ 2mN = 3 ANmN-, + 2mN = cN . mNPl

Here y i and y k are prescribed values of the spline second derivative at the ends of the interval. Moreover, these two sets are equivalent provided

It is readily verified that the right-hand members of (2.1.7) and (2.1.15) are three times the values of the second and first derivatives at xi, respectively, of the parabola through (xipl,y i P l ),( x i ,yj), and (xi+l ,y j t l ) having vertical axis. This fact has special significance relative to the problem of curve fitting with higher-order splines and is examined in detail in Chapter IV. A very efficient algorithm is available for solving the system of equations (2.1.8) or (2.1.16). Given the equations

+ + bZx2 + + + + + b,x,

alxl

~ 3 x 2

an-1xn-2

b3x3

bn-lXn-1

anxn-1

form ( K

=

1, 2, ..., a )

c1x2

= dl

~ 2 x 3=

4

, 1

~ 3 x 4= d3 >

...

Cn-lxn = 4 - 1

+

bnxn =

dn

9

( 2 .I .20)

2.1.

15

INTRODUCTION

Successive elimination of x1 , x2 ,..., xnP1 from 2nd, 3rd ,...,nth equations yields the equivalent equation system Xk = qkxk+l xn = U n

+ Uk

(k

=

1, ..., It

-

l),

(2.1.21)

7

whence x,, x , _ ~ ,..., x, are successively evaluated. For matrices with dominant main diagonil, with which we are primarily concerned, this procedure is stable in the sense that errors rapidly damp out (0 < ck/pk < 1). We note also that the quantities p , and qk in the application to the spline depend upon the mesh A but not upon the ordinates at the mesh locations. Thus, several spline constructions on the same mesh may be carried out with the computation of only one set of p,’s and q,’s. An extension of this procedure is used for the periodic spline. For the equations

an-1~n-2

+ + + + + + + + bixi

~1x2

aixn =

~2x1

b2x2

~ 2 x 3=

bn-lxn-1

cnx1

...

4, 4 9

cn-ixn = 4 - 1

anxn-1

bnxn =

dn

9

9

we effectively solve for x, ,..., x,-~ in terms of x, by means of the first n - 1 equations and then determine x, from the last equation. In addition to the quantities p , , qk , ukdetermined previously, we therefore calculate for k = 1, 2, ..., n the quantities Sk

=

(so =

-Wk-l/pk

1)

(2.1.22)

so that Eqs. (2.1.21) are replaced by the relations Xk = qkxk+l

+

SkXn

+

01,

+ Uk

(k

=

1 , ..., n - 1).

If we write the equation Xk = t k X ,

then we have tk

= qktk+l

vk

= qkvk+l

(k = 1 , 2,..., n

+ +

-

(2.1.23)

I),

I), ( % = 0). (&

sk

uk

=

We determine tkpl ,..., t , , vkPl ,..., v, and evaluate x, from the equation cn(t1xn

+ + 01)

an(tn-ixn

+

wn-1)

We then determine xnP1 ,..., x1 from (2.1.23).

+

bnxn =

4

*

11.

16

THE CUBIC SPLINE

We note that, when a k , b,, and c, are all constant, the quantities and q, can be obtained from a solution of a second-order difference equation. Set p , = h,/hkp, with h, taken as 1. Then, with a, = a, b, = 6 , and c k = c for all A, we have from (2.1.20) the relations

pk

Pk =

aqk-l

+ b,

q k = -c/pk

7

so that there results the difference equation

h, - bhk-1

+

achk-2 = 0.

A similar property holds for the periodic spline. 2.2. Existence, Uniqueness, and Best Approximation For most cases of interest, the proof of the existence of the spline function involves merely an application of Gershgorin’s theorem(cf.Todd [1962, p. 227]), which states that the eigenvalues of the matrix (ai,j) (i,j = 1, 2,..., n) lie in the union of the circles

1z

- aii

I

=

1 1 a i , j1

j#i

(1

< i < n)

in the complex plane. A matrix with dominant main diagonal (1 aii 1 > CiTi I ai,j 1) is nonsingular. I n (2.1.9) and (2.1.17), the sum 1 a i . j 1 is always equal to 1, with aii = 2. I n (2.1.8) and (2.1.16), the condition for dominant main diagonal is that A,, p, , po , A, be less than 2 in absolute value. Thus it is seen that the periodic cubic spline with prescribed ordinates at mesh points always exists and is unique, the representation being given by (2.1.2) with the M iuniquely determined by (2.1.9), and that the same is true in particular for nonperiodic splines having cantilevered ends (m, and m , prescribed), having simple end support ( M , = 0, M , = 0), having prescribed end moments, or having simple supports at points beyond the mesh extremities (e.g., Mo = AM, , M N = p M N - l , 0 < A < l , o < p < 1). A general existence theorem covering a much wider class of nonperiodic cubic splines is given in Section 2.9. There it will be necessary to prove special properties of the coefficient matrix. We remark that more than one spline may be associated with a set of values for the quantities M i. Replacing yi by yi + mxj + C for fixed m and C does not affect the right-hand member of (2.1.7). Boundary conditions (2.1.18) possess the same property. Thus, s,( Y ; x ) mx C =

+

+

2.2,

17

EXISTENCE, UNIQUENESS, AND BEST APPROXIMATION

+

+

Sd(P; x ) , where Ti= yi mxj C. For the periodic case, we may only say that S,(Y; x ) C = Sd(P;x), with pi = yj C. ( T h e only periodic linear function is a constant.) A related question concerns the arbitrariness of the quantities M ior m i . Is there always some spline associated with an arbitrarily prescribed set of values of M ior mi ? It is seen for periodic splines that adding the corresponding members of (2.1.5) ( j = 1, 2, ..., N ) gives the necessary condition

+

+

c (4 + h,+l)W N

?=I

= 0.

It may be seen, however, that any set of values of the M isatisfying this relationship is an admissible set. If we designate the left-hand member = 0. Set of (2.1.5) by $j, then the preceding equation implies (yl -YN)/h1 C. Then(yi -yj+l)/hj = c $1 ... $j-1 ( j = 2,..., N ) . $N , but T h e equation system requires (yl - yN)/hl = c $1 this is equal to c. We now have the relations

+ + +

x:l$i

+ + +

1

Y1 Y2

YN

+ hc, = Y1 + h2(c + $11, ... = + + + +

=YN

YN-1

hN(c

$1

' I '

4N-l).

These equations have a one-parameter family of solutions (parameter yN) ,[: hi($l ... $i-l)]/(a - b ) , which is a known iff we take c = C function of the given Mi's. It is readily seen for nonperiodic splines that there is no restriction upon the quantities M i. T h e corresponding problem for the slopes mi is somewhat more complex. If we designate by 3z,hj the left-hand member of (2.1.14) and set (yl - yN)/h12= c, Eqs. (2.1.14) ( j= 1,..., N ) require that the following equations be satisfied:

+ +

(Yz -Y1)lhz2 = 41 - c, (YB - Yz)/k2 = $2 - $1

...

(YN

-yN-l)lhN2

(Yl -yN)/h12

= #N-1 #N

+ c,

- #N-2 - #N-1

+ f (-l)N-z#l + (-l)N-lc, + ''. + (-l)"'h + (-l)Nc*

These are consistent iff z,hN - z,hNP1 This is equivalent to the condition N

i=l

ml

*'*

+ *.. + (-1)"-'

+ h,

mN

[ I - (-l)"]

z,hl

+ (-l)Nc

= c[l

-

= C.

(-1)N].

18

11.

THE CUBIC SPLINE

T h e resulting system of equations has a one-parameter family of solutions for (yl ,..., y N )iff in addition C[h12- hz2

+ + (--I)N-’hNZ]= -[hZ2#, + h3’(1CI2 - + **.

+

#1)

hN2(#N-l

- #N-2

**.

+ + ”*

(-1)N-2$1)]*

We restrict our attention here to the case in which the intervals are of equal length. If N is odd, these requirements are equivalent to #z .-* # N = 0 and c = $ N . If N is even, there $3 exists a two-parameter family of solutions y z j = y N+ h2($, + + $2j-l), ... $zj) (parameters y N and c) iff y2j+l= y N 15% hz($2 $zN-l = $, $2N = 0, that is, iff

+ + + + + + + + + + + +...+ m1

+ m3 + ... + mN-1

=

m2

+

+ m4 + .*. + mN = 0.

Equations (2.1.7) or, alternatively, (2.1.14), are given added significance if we consider the following extremal problem. Let f ”(x) be continuous. For a given mesh A , let fj = f(xj) and let Sd(f;x) denote the periodic spline of interpolation to f(x) or, alternatively, the nonperiodic spline satisfying end conditions (2.1.18i). Thus, Sd(f;xj) = fj . Let Sd(x) be any cubic spline on A . Form the integral E

=

Sb

[ f ” ( x ) - S:(x)lz dx.

a

T h e quantity E is, of course, a measure of the approximation of S,”(x) to f”(x) on [a, b]. Let Mj denote &’,”(xi). Expanding the integral and integrating by parts gives E the form

+ c h3. (Mj2_,+ Mj-lMj + M 2 ) ,

(2.2.1)

_1

j=1

wheref; = fh,fo=f N , M o=M,iff(x)andS,(x)haveperiodb-a.The function E has a stationary point in the nonperiodic case when the conditions

hj -3

(L+l- f j hii-1

+3hj+l M j + hy Mj+l -

fi

hj

) \1

-

0 (j = 1,2,...)N - I),

(2.2.2)

2.3.

CONVERGENCE

19

are satisfied. I n the periodic case, a stationary point exists when the second relation is valid for j = 1, 2, ..., N . These conditions are equivalent t o (2.1.8) with boundary conditions (2.1.18i) or to (2.1.9). Consequently, the function E has a stationary point among the various choices of M j iff Sd(x) = Sd(f ; x). We shall show that this stationary point is actually a minimum point. Denote by ( M o ,Ml,..., M N ) and (M1 ,..., M N ) the solutions of these equation systems for the nonperiodic and periodic cases, respectively; that is, Mi= Sl(f ; x,). We rearrange the expression for E as follows: multiply the algebraic expressions in Eqs. (2.2.2), with Mj in place of M i, by -2M0, -2M1 ,..., -2MN, respectively, and add to the right-hand side of (2.2.1). We obtain expressions for E in the forms

- M j B i P l - 2MjMi] =

=

j”[f”(x)]’ dx + a

h.

j=1

j”[ f “ ( x > l 2dx + j.” a

$ [(Mj-l - fli-d2+ ( M j - , - I@-l)(Mj - Mi)

a

[S;(x)

.)I2

- S;(f;

dx

-

Observe that the first and third terms in the last member are independent of the choice of M i .It is evident, therefore, that E is minimum for Si(x) = Si(f;x). This is the best approximation property of the spline interpolation. I n Chapter 111, this and other related extremal properties are explored by more elegant and powerful means.

2.3. Convergence The effectiveness of the spline in approximation can be explained to a considerable extent by its striking convergence properties. If f ( * ) ( x )is continuous on [a, b] ( q = 0, 1, 2, 3, or 4), we find that Sd(f;x) converges tof(x) on a sequence of meshes at least as rapidly as the approach to zero of the qth power of the mesh norm 11 A 11 = maxjhj. (To compare with the degree of convergence of more general approximating sequences, see Davis [1963, Chapter XIII].) Similarly, SLp)(f;x) converges to f ( ” ) ( x ) (0 p q) at least as rapidly as the ( q - p)th power of the

< <

20

11.

THE CUBIC SPLINE

mesh norm. I n some cases, it is required that the ratio of maximum interval length to minimum interval length in the respective meshes be bounded, but for many cases of interest it is required only that the limit of the mesh norm be zero. These rates, moreover, are optimal. For purposes of demonstrating convergence, it is necessary to have at hand certain properties of the inverses of the coefficient matrices in (2.1.8), (2.1.9), or, alternatively, (2.1.16), (2.1.17). For the situations in which A,, , pN or A,, po are less than 2 in magnitude, the convergence proofs are relatively simple. I n this section, we restrict our attention to such situations, postponing to Section2.9 the derivation of the more incisive properties of the inverse matrix essential to the general case. If B is an 7t x n matrix and we take the norm on the space of n-tuples x = (xl , x2 ,..., x,) to be the sup norm,

then the induced norm on the matrix B (cf. Taylor [1958, Chapter 1111) is the row-max norm. T o see this, express the linear transformation associated with B as

c n

ye

=

bijXj

j=1

(1

< i < n).

Thus the induced norm B satisfies

Taking i* to be the index of a row yielding a maximum value for C;', I bij 1 and setting xi = 1 if b,*j 3 0, -1 if b,*j < 0, gives here the relation

It follows that we have the equation

2.3.

21

CONVERGENCE

Assume now that the main diagonal of B is dominant. For a given x, choose k so that 11 x 11 = I xk 1. Then

11 Y 11

=

Since B-I exists and x

I1 Bx /I = mfx

=

B-ly, we obtain the bound on 11 B-1 11:

Denote by A and 2 the coefficient matrices in (2.1.8) and (2.1.9), respectiv.ely. I n (2.1.8), assume that A,, and p N are numerically less than 2. Then in t h e nonperiodic case we have the inequality

11 A-l /I d max[(2 - A 0 ) - l , (2 - pN)-I, 11.

(2.3.1)

For the periodic spline, we have

I1 A-l /I < 1.

(2.3.2)

Let B and B denote the coefficient matrices in (2.1.16) and (2.1.17), with 1 po I < 2 and 1 A, I < 2 in (2.1.16). Then we obtain

II ~ - II1 G m 4 2 - p o ) - l , (2 - ~ ~ ) - 1 , 1 1 , 11 8 - 1 11 G 1.

(2.3.3) (2.3.4)

Now take {A,} to be a sequence of meshes on [a, b ] : d k

:

U

< X k , l < ". < X k , , r k

Xk,o

= b.

(2.3.5)

Set hk,j

= xk,j

- xk,i-l

and define the norm of A , to be

11

11

=

lsyNk

(hk,j)*

(2.3.6)

22

11.

THE CUBIC SPLINE

We shall be concerned with sequences {A,} for which 11 A , (1 --t 0 as k + 00. Under certain circumstances, we shall also require the additional restriction on the meshes d, that

(2.3.7)

I n particular, we require this restriction in the first convergence theorem:

Theorem 2.3.1. Let f ( x ) be continuous on [a, b]. Let {A,} be a sequence of meshes on [a, b] with lirnk+a,11 A , I j = 0 and satisfying (2.3.7). If SAk(x)is the periodic spline of interpolation to f ( x ) on A , , or the nonperiodic spline of interpolation satisfying end conditions (2.1.18iii) with max[i Ak,O 1, 1 P k . N , 11 < and 11 11'( 1 dk,O 1 I d k , N k 1) - 0 as k -+ CQ, then we have uniformly with respect to x in [a, b]

+

f(%) - s A x ( X )

11)

=

(2.3.8)

If, in addition, f ( x ) satisJies a Holder condition on [a, b] of order a(0 < 01 < l), then

f(%)

- sA,(X)

=

O(I1 '

uniformly with respect to x in [a, b], provided is bounded in the nonperiodic case.

Proof.

k

Ila) 11 d, 112- u ( 1

(2.3.9)

dk,OI

+ 1 dk.N, 1)

On [xj-l , xj], we have from (2.1.2) the relation

where, for simplicity, we have dropped the mesh index k. If A,: are the elements of the inverse of the coefficient matrix A in (2.1.8), then we find

Let p( f;S) be the modulus of continuity of f ( x ) on [a, b]. We note that the coefficients of Mi-1 and M j in (2.3.10) do not exceed hj2/35J2 in magnitude. Also by (2.3.7) follows the inequality

2.3.

23

CONVERGENCE

By (2.3.2), therefore, we have h?(1

Mj-l

1

+1

1)

Mj

< 11 A-1 li{jBzdf;11 A 11) + / / A lI'(1

I

+1

dN

1) .

(2*3.12)

I n the periodic case, we have simply

V(IMj-1 I

+I

M j

I)

= 11

A-1 Il$P2& I1 A 11)

< QB"(f;

II A 11).

Thus we obtain from (2.3.10), in the nonperiodic case, the inequality

I

s A ( x ) -f(x)

I

< [1/35'21II A-l ll{QBz~(f;/I A Ill + /I A llz(l + Af;II A ll)(l + 4).

do

I

+ I d N I)>

A corresponding inequality is evidently valid in the periodic case, and (2.3.8) follows. 1) If, furthermore,f(x) satisfies a Holder condition of order cy(0 < 01 on [a, 61, that is, if for some constant K

<

If@>

-f(x')

I
- x'

la

for x and x' on [a, b], then the right-hand member of (2.3.1 1) is replaced by the expression 4P2K/I A /Ia:) and the inequality

I sA(x)-f(x) I

< [1135/21 /I

~

-

II1 I/ A II"{QK(B~ + 1) *

+ I/ A I I ~ - ~do(I/+ I

results. I n the periodic case, we have

I

S A ( ~-f(x) )

i

< [1/3"/"1/I k1I1 /I A Ila *

*

+ 1).

QK(Bz

This completes the proof of the theorem. I t is evident that the corresponding result for end conditions (2.1.19iii) is valid. Here we make the restriction that sup,(max[/ p k , f lI, 1 A k , N , I]} < 2; furthermore, 1 C k , N , 1) + O as k-+ 00 for (2.3.8) and that we require that /I 11(1 ck,fl I 11 A , 111-=(1 ck fl I I c ~ , I)~ be , bounded with respect to k for (2.3.9). It is worth noting at this point a connection with certain properties of polynomials investigated by FejCr. FejCr [I9301 showed, for a polynomial g(x) of degree 2n - 1 which satisfies at the Tchebycheff abscissas 5.p) (k = 1, 2, ..., n) on [-1, 13 the inequalities I g(tp))l A and ]g'(.$in))i B , that on [-1, 11, 1 g(x)J A A,B, where A, is independent of the polynomialg and A, + 0 as n + 00. He had previously demonstrated [1916] that if a polynomial X,(x) of degree 2n - 1 interpolates to a functionf(x) of C[-I, 11 at these Tchebycheff abscissas with X;(&,)) = 0, then X,(x) converges uniformly tof(x) on [-1, 13.

+

<

+

< +

<

24

11.

THE CUBIC SPLINE

Consider here the representation (2.1.10) for a spline Sd(x) which is valid even if S,(x) is not in C2[a,b]. We note that, if we set u = ( x - xj-l)/hj on [xjPl,4, we have S&)

= m,-,h3(

1 - u)% - rnjhjUZ( 1

+yp2[2(1

Hence, if 1 Yk

1

- 0)

+ 11.

- a)

+ yj-,( 1 - a)2(2u + 1)

< A and I mk 1 < B for k = 0, 1,..., N , then on SA(4

5 A + (W4)B.

, xj]

Thus we obtain the following lemma.

Lemma 2.3.1. If a cubic spline Sd(x) on an arbitrary mesh d on [a, b] satisfies the inequalities

then we have on [a, b]

It is important to remark that in this lemma S,(x) is a piecewise cubic with continuous first derivatives. No assumptions have been made on Si(x). Using terminology that will be developed more completely in subsequent chapters, we speak of such splines as being of dejiciency 2, splines with continuous second derivatives generally as being of dejiciency 1. A spline that is a single cubic arc we designate as being of dejiciency 0. We note next that if we employ splines of deficiency 2 which interpolate to a continuous function f ( x ) at points of a mesh d on [a, 61 and at these points have derivative 0, we obtain the analog of FejCr’s result of 1916: Lemma 2.3.2. Let {A,) be a sequence of meshes on . [a, b] with 11 A , 11 -+ o as k --+ 00. Let f ( x > be of class ~ [ ab]. , Let SAk(x)be the spline of dejiciency 2 interpolating to f ( x ) at the points of A , and there having derivative 0. Then the sequence {$,(x)) converges uniform5 to f ( x ) on [a, b]. In f a c t ,

>.(fI

- %k(4 I

< 2 d f ; I1

IP).

Proof. T h e existence of such splines is evident: at the extremities of each interval of A , , we have prescribed the value of a cubic and its

2.3.

25

CONVERGENCE

first derivative. Next, the spline sAr(x) is monotone on each interval of A , . Thus on [ x , , ~ - ,~x,,~] we have

T h e following lemma involves somewhat more relaxed requirements on the spline involved. Its proof is evident.

Lemma 2.3.3. Let {A,} be a sequence of meshes on [a, b ] , with

/I A , 11 -+ 0 as k + co. Let $,(x) and pA,(x) be two splines on A , which coincide at the mesh locations. If muxi j s L , ( ~ ~ ,11~A),l Ij -+ 0 and maxi 1 F~(X,,~)I /I A , /I 0 as k 00, then on [a, b] --+

--+

sA&)

-

t&) = o(1)

uniformly with respect to x in [a, b]. Let us now return to Theorem 2.3.1. With no additional mesh restriction, we have the splines sA,(x) of deficiency 2 of Lemma 2.3.2 converging uniformly to f(x) on [a, b]. With the additional mesh restriction (2.3.7), we find for the splines SA,(x) of the theorem, using the boundedness with respect to k of 11 A-l /I and 11 A-’ 11 in (2.1.16) and (2.1.17), that maxj

I sx.k,j)

I 11 A , I1

-+

0.

Thus, by Lemma 2.3.3, [SA,(x) - sAk(x)]-O, and by Lemma 2.3.2, SA,(x)-+f(x), both uniformly with respect to x in [a, b]. This alternative proof of Theorem 2.3.1 indicates somewhat more clearly the role of the mesh restriction (2.3.7). If we assume thatf’(x) is continuous on [a, b], then the spline and its derivative converge. Furthermore we no longer require the condition (2.3.7) on the sequence of meshes to be satisfied.

Theorem 2.3.2. Let {A,} be a sequence of meshes on [a, b] with limk+oallA , 11 = 0. L e t f ( x ) be of class C’[a, b].Let the splines ofinterpolation SA,(x) satisfy end condition (2.1.19i)or be periodic i f f(x) is periodic. Then we have, uniformly with respect to x in [a, b],

11.

26

THE CUBIC SPLINE

If f‘(x) satisjies a Holder condition on [a, b] of order a(0 < ci (uniformly with respect to x in [a, b ] ) [f ( n ) ( x )- Si;)(x)]

=

O(il A , illfa-p)

(p = 0, 1).

< l), then (2.3.14)

Proof, Let mk = (m,,n , m,,l w,,N,)~, c k = (C,,n c,,l 7 ’ . . , c k , N k I T , f , = ( fk,o, fk,l ,...,f,,Nk)Tin the nonperiodic case, with corresponding expressions for the periodic case. For a given A , , let B, denote the coefficient matrix in (2.1.17), or (2.1.16) with end condition (i) and the first and last equations written as 3mk,, = 3 f ‘ ( a ) = c , , ~ , 3mk,Nk= 3f’(b) = c , , ~ ,. Then 3

j...?

Bk(mk

- ck/3)

= (‘k

- Bk/3)ck

I

+

+

where I, is the ( N , 1) x ( N , 1) or N , x N k unit matrix. T h e right-hand member is, accordingly, in the nonperiodic and periodic cases,

T h e norm of neither of these vectors exceeds p(f’; 3 ]Idk11) < 3 p ( f ’ ; 11 A , ll), and so I/ mk - c k I1 d II Bil II * 3 4 . f ’ ; II A , 11) d 3 4 f ’ ;/I A , 11). Now c,,? = f ’ ( [ l c , j ) at some location f k , j in [ x , , ~ - ~x, ~ , in ~ ]the periodic case, and the same is true for 1 < j Nk - 1 in the nonperiodic situation. Of course, c , , ~= 3f’(a), and c , , ~ , = 3f’(b) in the latter instance. Thus, we obtain I/ mk - f; I/ d 4p(f’; /I A , 11) . As a result, we find for x on [ x , , ~ - ~x, ~ , that ~ ] we have from (2.1.11) (dropping the mesh index k)

<

2.3.

27

CONVERGENCE

< p ( f ‘ ; hj), we

Thus, since / f ’ ( x )- (fj - fjPl)/hi/ equality

obtain the in-

I f W - S ’ ( 4 I < M f ’ ; /I A 11).

If I f ’ ( x ) - f ’ ( x ’ ) l

< Kl x - x’

on [a, b], then we see immediately that

I f ’ ( 4- S g X ) I

< w I/

Il=.

Because of the interpolation property, we have on [xi-l, xi] the relation

Equations (2.3.13) and (2.3.14) now follow. The case of the general end condition (2.1.19;;;) will be considered in Section 2.9. Essentially the same method of proof applies to the situation in which f”(x) is continuous on [a, b]. The following theorem extends somewhat the results of Sharma and Meir [abs. 1964; 19661 by admitting more general conditions on f ( x ) .

Theorem 2.3.3. Let f ( x ) be of class C2[a,b]. Let (A,} be a sequence of meshes on [a, b] with limk+m/ j A , 11 = 0. If the splines of interpolation SdLx) to f ( x ) on A , satisfy end conditions (2.1.18i) or (2.1.18ii), or i f Sd,(x) and f ( x ) are periodic, then we have = 0(l1

A , 1)”’

( p = 0, I , 2),

(2.3.15)

unifoymly with respect to x in [a, b]. I f f “(x) satisjies a Holder condition on [a, b] of order 01 (0 < 01 I), then

<

[ff‘”’(x)- S~:)(X)] = O(ll A,

1 2+6-”)

( p = 0, 1,2)

(2.3.16)

uniformly with respect to x in [a, b]. Proof. We set M,= (M,,o, M,,l ,..., Mk,NJT,dk = (d,,o, d,,l,-.-, 4 , ~ in the nonperiodic case and (M,,l ,..., M , N k ) T , (dk,l ,..., dk,N,)T in the periodic case. Let A , denote the coefficient matrix in (2.1.8) or (2.1.9) associated with the mesh A , . Then (2.1.8) for end condition (i) and (2.1.9) can be written in the form - dk/3)

Z=

(Ik

- Ak/3)dk

>

(2.3.17)

~ )

28

11.

THE CUBIC SPLINE

where Ik is a unit matrix. T h e right-hand member, in these two cases, takes the forms (dropping the mesh index k)

We note that d,/6 is the divided difference (Hildebrand [1956, p. 381) f[xj-l, xj , xj+J in the periodic case, and when j # 0 or N is the nonperiodic case,

[,

(,) at some location in the interval xi-1 < x < This is equal to if”( It is also true by the Taylor theorem that quantity d0/6,

.

if”([,)

is equal to at some location in the first interval, and a similar assertion can be made for dN . It fOllOWS, therefore, that ll(Ik - Ak/3) dk 11 311 dk 11) 3 p ( f ” ; 11 dk \I), and we obtain from (2.3.17)the inequality

< Y(f”;

< Ilfi dk/311 < p ( f ” ; 11 dk II), and it follows that 11 < 4p(f”; 11 d k 11). From the piecewise linearity of Sik(x),we

since I( Ail 11 1. It is clear that

11 M k - f

<

-

obtain the inequality

If f“(x) satisfies a Holder condition of order for some K independent of k and x

CL

on [a,b], then we find

2.3.

29

CONVERGENCE

I n consequence of the interpolation property of SAk(x>, an application of Rolle’s theorem yields the fact that in every interval [ x ~ , ~x- ~ , ~ ] there exists a point f k , j for which f ’ ( f k , j = ) s L A f k , j ) . Thus on this interval we find that

and a second integration yields the property

Relations (2.3.16) and (2.3.17) are an immediate consequence. The nonperiodic case involving end condition (2.1.18;;) is included by writing the first and last of Eqs. (2.1.8) as 3Mk,, = 3 f ” ( a ) = dk,o and 3Mk,N, = 3f”(b) = dk,NI, . We still have 11 Azl / j 1. T h e convergence properties become even more striking as we increase the smoothness of f(x). Birkhoff and deBoor [1964] showed that, for the nonperiodic splines of interpolation satisfying end condition (i), if f”(x) is absolutely continuous in [a, b], then Szk(x) converges uniformly tof”(x) provided the mesh condition (2.3.7) is satisfied. Moreover, if fi”(x) is continuous, then f(p)(x) - Si:)(x) is of order O(ll A k ]I4- ”) ( p = 0, I, ..., 4) uniformly with respect to x in [ a , b]. We present here a somewhat simpler proof of slightly stronger forms of these theorems.

<

Theorem 2.3.4. Let f(x) be of class C3[a, 61. Let {Ak} be a sequence of meshes on [ a , b] with limk+m11 dk /I = 0 and satisfying (2.3.7). Then for the splines of interpolation S,Ax)satisfying end condition (2.1.18) ( i ) or (a), orperiodic if f(x) is periodic, we have, uniformly with respect to x in [a, b], [ f ‘ ” ) ( ~-) S ; ~ X )=] o(II A, l i 3 - ” )

( p = 0, 1, 2, 3)

If f”‘(x) satisJies a Holder condition on [a, b] of order

cy.

(0 < cy. < I), then

uniformly with respect to x in [ a , b]. Proof.

Consider, first, Eqs. (2.1.8) with A.

(2.3.18)

=pN=

1 . Set

30

11.

THE CUBIC S P L I N E

the third derivative of the spline in [xj-, , xj]. In (2.1A), subtract from each equation but the first its predecessor, obtaining

+

+

4 PlhlOl + ( 1 + Pl + XZ)h2% + @ 3 u 3 = 4 - 4 > PZh2% (1 + P2 + h 3 ) h , U 3 + h 3 h 4 0 4 = d 3 - 4 (1

~N-zhN-2uN-z

+ (1 +

+. . .

pN-2

+

h1)hlUl

+

= dl -

9

2

+ + (1 +

hN-1)hN-iuN-i

~N-ihN-iuN-1

AlhZU2

(2.3.20)

AN-ihNuN

11.N-i)hNuN

+ +

=

dN-i - dN--2

= dN -

9

d ~ - *i

+ +

Divide through by (h, hz), (h, h2 h3), (hz h3 k),..., (hN-z+ hN-, + h N ) , (hN-l h N ) . T h e right-hand members become the column vector Y = ( r , , y2 ,..., r N ) = ,where

+

Y,

= 6[f,-2

!f,-1

, f , ,f,+ll

( j = 2,...,

-

1).

Here [fj 7fj+l,...,fj+J = f[xi , xj+, ,..., xj+J denotes the kth divided difference of f(x) on the points xi, xj+, ,..., Note that the following relations are valid:

T hus we obtain from (2.3.20) the set of equations

...

(2.3.21)

2

PN-ZPN-I - PN-AN-1

uN-2

P;-luN-1

+ (l +

PN-l)’N-lUN

= ‘N

*

Although the coefficient matrix here collapses for A, = = AN-l = 8 to a dominant-diagonal matrix, the main diagonal is not generally

2.3.

31

CONVERGENCE

dominant. T h e coefficient matrix C , however, can be represented by a matrix possessing this property multiplied by two diagonal matrices: C = EDF,

where '1

0-

Pl PlPZ

1

A1 -

- PlAZ

Pl

PlP2

F=

E=

0

1

+ A, P1

D = [

1

+ + Pl

...

0

A2

1

PZ

+ Pz + ... 0

...

0

... ... ...

0 0

0

A1

A3

A3

0

0

It is clear that I/ D-l < 1. If now, for all i , j , 'the-intervalsof the mesh satisfy the restriction hi/hj

< B < a,

then we have the inequality

Furthermore, we see that

1

1

+p

Thus it follows that

< .<-

'p.3'

1+B

<1

II c-' I1 < (1

(l<j
+ 2B)B2.

1.

11.

32 Let o = (a, ,..., u C(U

- 7) =

1 - Pl(1

-Pl2P2

~ )and ~ , from

Ca

r obtain the equation

=

( I - C)? =

+ A,)

(1 - PlA2)

THE CUBIC SPLINE

0

--Xi2

1

(1

+ Pl + A2)AlPZ 1

- PlA2

--1AZ2

...

1 - PlA2

0

0

0

...

0

0

...

0

0

...

-Pk1

1

1.

- hN-1(1

f

PN-1)

T h e sum of elements in each row is zero. T h u s the last member is the vector

I -

-h2(r2 -~X J Z 2 1 -P A

(73 - 7 2 )

+

P12P2 ~

...

kzA;-l

1

- T1)

(TN - 7 N - i ) - P N - ~ N - ~

Pk-l(rN

+

1

- PJ2

(72

- 71)

2

PN-ZPN-1

(%-I

1 - PN-JN-I

- IN-2)

- IN-1)

Iff”’(%)is continuous on [a,b ] , then 1 rjil - r j I can be made arbitrarily small. Since (1 - pjAj+l)-l < 1 + 3/3, the step function ST(x) can be made uniformly close tof’”(x) by taking 11 d 11 sufficiently small. In fact, I rj+l - rj I G p ( f m ; 411 11) G 4 p ~ f ’ ” ;11 IN, and

I1 I 3

-r

I/ < KZCL(fl7 /I A Ill,

+

38) 4. Since on [xj-, , xi] we have where K , = (1 + 2/3)P2 * 2(1 If”’(x) - r j I p(f”’, 311 d I[), it follows that

<

I Sl;(X)

-f”’(X)

I

< ( 3 + KZMfm, I1 All).

Iff”(x) satisfies a Holder condition of order a, 0

I Sp)-f”’(X) I

= O(lI A

(2.3.22)

< a \< 1 , on [a, b ] , then Ila)

uniformly with respect to x on [a,b ] . I n particular, iffi”(x) is continuous on [a,b ] , then

I q ( X ) -f”’(x) 1

< ( 3 + Kz)

Ifi”(x) I

*

/I A /I.

Only minor modifications are required for end condition (ii). T h e first and last equations of (2.1.8) are written as 3M0 = do = 3f”(a) and 3 M N = d, = 3f“(b). T h e quantities 2 A, and 2 p N - l replace

+

+

2.3.

33

CONVERGENCE

1 + A, and 1 + pN-l in the first and last lines of (2.3.20),(2.3.21),and the matrix D. T h e periodic case requires a slight change in the handling of the coefficient matrices. I n this case C = EDF, where

E=

F=

0 0 0 0 0

0 0

...

...

PN-2

1

+ PN--2 +

0

PN-I

PN

0 0 AN-i

1

+ PN-1 +

Ahr

I n this, we have used the fact that, in the periodic case, X,X, ... AN = plpz pN . T h e bound on /I C-l(I is as before, and we obtain again

(2.3.22).

T h e third derivative, S i ( x ) , however, has jump discontinuities at mesh points so that simple integration cannot be employed to obtain inequalities on the second derivatives. T o study the second derivative, then, we note the relation for second divided differences: f. 3tl

-f.

h+l

1 -

hj

f. - f. 3 3-1

+ hi+l

1

hj+l

sx,

+-

‘3fl

2 -

t>2f“’(t) dt

2(hj

1

-hj

1

xJ x5-1

-

t>2f”’(t) dt

+ h+l>

Using (2.1.5) and the interpolation property of SA(x), we obtain f y -M. 7

3 -

34

11.

THE CUBIC SPLINE

On [xi-1,xi], we evidently have f”(x) - S i ( X ) = f ; - M j

+

JZ

[f”’(t)-

dt,

2 7

and we obtain from this relation the inequality

G Q I1

II (3

+ Kz)CL(f”’;/I A ll),

where we have made use of (2.3.22). The corresponding properties for Sb(x)- f ’ ( x ) and Sd(x) -f(x) now follow by integration and the application of Rolle’s theorem as before. Two further properties serve to complete this initial presentation of spline convergence properties. We only mention these here and postpone their proofs to Sections 2.9 and 3.12. Iffiw(.) is continuous on [a, b] and if {A,} has the properties 11 Akll 0 < co, and maxi I A,,j - I + 0, then as k + co, /I A , ll/minjh,,j ---f

<

lim max k+m

-sz(Xk,j-)

SJk(xk,j+)

- f””(Xk.j)

/ / A k 11

3

2

I =o.

Thus the jumps in the spline third derivative are tied directly to the fourth derivative of the approximated function. Finally, the uniform rate of convergence of SL:’(x) - f ( p ) ( x ) to zero was shown to be o(ll A , 1Iy-p) when f(x) is of class CY[a,b] ( y = 0, 1 , 2, 3, 4;0 < p y ) and to be O(1l A , 1I4-p) when f ( x ) is of class C4[ a,b] (0 < p 3). I n Section 3.12 it will be shown that it may be no higher than O(ll A , 1I4-p), unless f(x) is itself a cubic polynomial.

<

<

2.4. Equal Intervals

I n the case in which the intervals of the mesh A on [a, b] are of equal length, the inversesof the coefficient matrices (2.1.8) and (2.1.9) or(2.1.16) and (2.1.17) take on relatively simple forms. These permit the immediate application of the spline to standard problems of numerical analysis. Introduce the n x n determinant 2 I-h Dn(4 =

h 2

...

h (2.4.1) 1-h

2 I-h

h

2

2.4.

35

EQUAL INTERVALS

This determinant, where DP1(A) = 0, Do(h) = 1, and Bl(A) = 2, satisfies the difference equation D,(h) - 2Dn-1(h)

+ h(1 - h)D,-S(h)

= 0,

so that

We set D,

=

D,(Q), Do = 1 , D-,= 0. Then we have D,

=

(1

+ 3’/’/2)”+l31/2 - ( 1 - 31’2/2)”+1

(2.4.3)

Define the n x n determinant

Qn(a)

Here Q1(ol) = 2. We define Qo(ol) = 1. For n

2 1,

Q n ( 4 = 20%-1- ( 4 W n - 2

9

(2.4.4)

and Q n ( o l ) satisfies the same difference equation as D, . T h e coefficient determinant I A 1 in (2.1.8) satisfies the equations

1A I

~ Q N ( P N) XOQN-~(PN)/~ = 2QdhO) - PNQN-l(X0)/2*

(2.4.5)

We obtain the elements of the inverse .A-l of the coefficient matrix in (2.1.8) in the case of equal intervals from the cofactors of the transpose matrix :

36

11.

THE CUBIC SPLINE

T h e inverse matrix A”-l for the periodic spline becomes particularly interesting if the intervals of d are of equal length. The matrix is now a circulant matrix (cf. Muir [1960]). Each row may be obtained by advancing each element of the preceding row to the next following position. Properties of such matrices play an important role in the development of polynomial splines and are examined in some detail in Chapter IV. For the determinant I A I of coefficients in the periodic case (2.2.6) with equal intervals, we obtain the relation

1A1

= 2DN-1

-

+ (-2)lPN.

ON42

(2.4.7)

Let dbN),d i N )..., , dAF\ be the elements of the first row of A-l, and adopt the convention dkT\ = djN)= d-( y ) . Then we have = d:?:.

(2.4.8)

T h e quantities d i N )are again found directly from the cofactors of the coefficient matrix, which is symmetric: dLN’= [(-2)-N+kD k-1

+

(0

A [

(-2)-kDN-k-J/l


-

1).

(2.4.9)

For Eqs. (2.1.16), the determinant 1 B I of the matrix satisfies the equations I B I = 2QN(PO) - ~NQN-l(P0)/2 = 2&N(hN) - POQN--I(XN)/2, (2.4.10) whereas for (2.1.17) we have B = A” and hence In the nonperiodic case, the moment Mi

N

=

2 A,:

IB1

=

1 A” I.

-

2K.t

dj

j-0

is seen to be given by

Mi

= A,@o

+

+3

l

A,:fo

+ (AT;

+

- 2Ac:-1)fN-1

(&k-Z

+ 1 (ALi-1 + N-2

- 2A;:)fl

j=2

ALb-lfN/

Azk-ldN

if we require the spline to assume the value j # i, we have the relations A:? a.7-1 A,:

+ - 2AL: + A,:

-

2A;i

ATk-2 - 2A;k-l

=

-64:

=

-6A;:

+ A<& ==

-6ALk-1

fj at

+ A<:+Jfj

9

xj = a + j h . For

(1 cj < N - l),

+ (1 - 2Ao)ALt( i f

+ ( 1 - 2pN)A;k

l), (i# N

- l),

2.4.

37

EQUAL INTERVALS

whereas

Thus we obtain the representation for the moments,

I n the periodic case, we obtain (2.4.12)

T h e quantities A<; and A;,: decay rapidly as j departs from i. Thus it is necessary in the evaluation of the quantities M j to employ only a few terms centered at xj . I n addition, the limiting values of the coefficients in (2.4.1 1) and (2.4.12) as N -+ co may frequently be used, affording considerable simplification in applications of the spline. Set Y =

1

+ 3lI2/2,

s = 1 - 3'" 12,

u =

-1/(2r)

=

-2s

= -(s/r)112;

then we find directly that D,

=

Using the fact that 2

p+l

- sn+l

r - s

'

1 - 02% Dn-1 Yn 2+u *

+ u = -(2 + l/u) = 31/2, we obtain the relation (2.4.13)

For the nonperiodic spline, those situations in which A, = p N = 01 are of particular interest. With 01 = 1 is associated the end condition (2.1.18i); with 01 = 0, (2.1.18ii); a = -2 gives a parabolic runout. This last case is related to the condition Mo = M I , MN-l = M N , which has special significance when neither the end slopes nor end moments are known. T h e existence and convergence properties of such splines will be covered in Section 2.9.

38

11. THE

CUBIC SPLINE

It is easily seen that, for n 2 1 ,

(2.4.14) Thus for 0

< i <j

< N a n d for i = j

= 0,

so that

A:? 2.3

=

&i[2 + olo - u2i-l (20 (2 4[(2

+

+

au)2

-o2y2a

+

u2N--2j--1

(20

a)2]

(0 < i < j

and

N-i[2 + ola - $-1 (2a 4(2 au)2 - o2N-2(2a + a)Z

AT1 z * N =(J

For0 < j


so that 0.j

- oj -

=

uj

*

(2.4.15b)

(0 < j < N),

(2.4.15~)

9

+

0.N

+

- (2

+

< N ) , (2.4.15a)

2aQN-i(~)/~N-j I A l/rN

2 4 2 au - u2N--23'--1(2a (2 + au)2 - o2N-2 (2u A-1 -

+41

(0 < i ,< N).

+

A;,:.

and

+ a)][2 + au -

+ a)] +

ON242 a) - u2N--2(2u

au)2

+

a)2 '

(2.4.15d)

To complete the evaluation of A,: , we remark that in consequence of (2.4.6) we have A;,; = A;,: (0 < i < N, 0 < j < N ) , whereas A<: = A;:t,N-j for unrestricted i and j . For the three cases referred to in the foregoing: End Condition (i) ( a = 1 )

2.4.

39

EQUAL INTERVALS

End Condition (ii) (a = 0)

A;,:

=0

(0 < i


(0 <j

d N).

Parabolic Runout ( a = -2)

(0 < i < N ) ,

T h e elements B;,: for equal spacing are obtained from expressions (2.4.6) by replacing A,, pN by p, , A,. Of course, B;; = A;.;. I n order to obtain the expression micorresponding to (2.4.1 l), we write

+

B
$-

'N)*

(2.4.16)

It is seen from the discussion following(2.1.19) that for the determination of the B;: from (2.4.15a-d) the values of ci in the three cases are 0, 1, 2, respectively. T h e first two are therefore available from the foregoing, whereas the case ci = 2 gives a parabolic runout:

40

11.

THE CUBIC SPLINE

Parabolic Runout (a = 2)

(0

< i < N),

I n the periodic situation, (2.4.16) is replaced by

(2.4.17)

where the brackets denote "integral part of." as N -+00 play a useful role. We have T h e limiting values of

T h e error in approximating A;! by dt'j is given by

Even in the nonperiodic case, this is a useful approximation except near the ends of the interval, inasmuch as CJ = -0.268. T h e values of dim' for k 7 are

<

k = 0 1 2 3 4 5 dLm' = 0.57735 -0.15470 0.04145 -0.01111 0.00298 -0.00080

of

6 7 0.00021 -0.00006

Table 2.4.1 presents the quantities dAN)that serve as the elements A'-': A';; = d j y i .

2.4.

41

EQUAL INTERVALS

TABLE 2.4.1 VALUESOF dkN”’,N

=

2,..., 17 ~

N=2

k

N=3

N=4

N=5

~

N = 6

N=7

N = 8

N = 9

0 1 2 3 4

0.66667 0.55556 0.58333 0.57576 0.57778 0.57724 0.57738 0.57734 -0.33333 -0.11111 -0.16667 -0.15152 -0.15556 -0.15447 -0.15476 -0.15468 0.08333 0.03030 0.04444 0.04065 0.04167 0.04139 -0.02222 -0.00813 -0.01190 -0.01089 0.00595 0.00218

k

N=lO

N=ll

N=12

N=13

0.57735 0.57735 0.57735 0.57735 -0.15470 -0.15470 -0.15470 -0.15470 0.04147 0.04145 0.04145 0.04145 -0.01116 -0.01109 -0.01111 -0.01111 0.003 19 0.00292 0.00299 0.00297 -0.00159 -0.00058 -0.00085 -0.00078 0.00043 0.00016 -

N=14

N=15

N=16

N=17

0.57735 -0.15470 0.04145 -0.01111 0.00298 -0.00080 0.00023 -0.00011 -

0.57735 0.57735 0.57735 -0.15470 -0.15470 -0.15470 0.04145 0.04145 0.04145 -0.01111 -0.01111 -0.01111 0.00298 0.00298 0.00298 -0.00080 -0.00080 -0.00080 0.00021 0.00021 0.00021 -0.oooO4 -0.oooO6 -0.oooO6 O.ooOo3 O.oooO1

T h e corresponding elements of the inverse matrix for the nonperiodic spline for the cases considered in Table 2.4.1 may be determined from the following identities, which are readily verified. For 01 = 1, ~12 . 3

=

dj!?)

+

A& = ; [ d p ;

For

01

=

(0 < i

A,:

+d

3

A&

=

2djZN) (0 <j


= ,EN).

0,

AT1 2.3 -

A&

<j < N ) , (0 < i < N ) ,

@)’

=

d3-22 Y ’ - dj!?)

(2 + 2

(0 < i < j

[,y,;f;

-


4qp]

A,:

=0

(0 < i < N ) .

(0 < j


42

11.

In order to obtain B,! for

01

THE CUBIC SPLINE =

2, we have

2.5. Approximate Differentiation and Integration Among the most important applications of the spline to numerical analysis and the areas where its versatility and flexibility are particularly evident are numerical differentiation and integration. From Theorem 2.3.4, it is evident that in approximate integration a fourth-order process results even for unequal intervals. For approximate differentiation, the error is O(h3),but there is more to add to this picture. T h e resulting derivatives at mesh points are “smooth”, a property attributable partly to the best approximation characteristic exhibited in Section 2.2 and partly to the minimum curvature property developed in Chapter 111. We have for the periodic spline, from Eqs. (2.1.15), the relations (2.5.1)

and, for the nonperiodic spline, the relations N-1

mi = B

+3 1

; ~ C ~

f’+l - f ’ ]

hj+l

,=1

+ B&c,.

(2.5.2)

For equal intervals, we have the formulas (2.4.15) and (2.4.16). I n the case of unequal intervals, however, it is rarely worthwhile to compute the quantities B;:. T h e algorithm at the end of Section 2.1 should be used to determine the slopes midirectly or to determine the moments M i whence the slopes may be found by (2.1.4):

A slight rearrangement of (2.5.2) and (2.5.3), however, yields an important property of the slopes m i .We have from (2.5.1) the equation N

mi=3C

,=I

fi

-ff-l

h,

[A&:

+ pj-lB;;-l],

(2.5.4)

2.5.

43

APPROXIMATE DIFFERENTIATION AND INTEGRATION

and from (2.5.2) we obtain the equation

(2.5.5) EXAMPLE 2.5.1 DETERMINATION OF SLOPES AND SECOND DERIVATIVES FOR A NOZZLE CONTOUR (ENDSLOPES PRESCRIBED) y” (spline-on-spline) X

Y

Y‘

M

0 I 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

5.160 5.110 5.070 5.020 4.972 4.921 4.860 4.738 4.528 4.228 3.829 3.373 3.094 2.985 3.100 3.375 3.720 4.050 4.380 4.700 5.000 5.280 5.550 5.580 6.050 6.275 6.500 6.700 6.890 7.070 7.250

-0.04400 -0.04450 -0.04780 -0.04924 -0.04923 -0.05083 -0.08343 -0.16442 -0.25486 -0.36231 -0.45763 -0,38834 -0.19400 0.00043 0.21077 0.32662 0.34276 0.32733 0.32793 0.31094 0.28830 0.27586 0.25826 0.25110 0.23733 0.22458 0.21433 0.19308 0.18334 0.18354 0.16250

-0.00490

0.00380 -0.0103 1 0.00743 -0.00741 0.00420 -0.06941 -0.09257 -0.08830 -0.09423 -0.12878 0.26736 0.12135 0.26724 0.15369 0.07801 -0.0457 1 0.01483 -0.01362 -0.02037 -0.02492 0.00004 -0.03524 0.02093 -0.04848 0.02299 -0.04349 0.00099 -0.02046 0.02085 -0.06292

(All

= PAr =

-0.00145 -0.00255 -0.00266 -0.00087 0.00186 -0.01135 -0.05907 -0.09313 -0.08270 -0.121 18 -0.04087 0.15803 0.19970 0.20913 0.17807 0.05750 -0.01 209 -0.00700 -0.00440 -0.01 245 -0.01 635 -0.01530 -0.01258 -0.00867 -0.01557 -0.00891 -0.01754 -0.0 1646 -0.00577 -0.00331 -0.00960

1)

44

11.

THE CUBIC SPLINE

I t is possible to show for equal intervals that the summation

in the sense of Schoenberg is a smoothing of the quantities ( fi - hPl)/hi [1946], and the calculation of mi involves an additional averaging of such sums. Related conclusions can be drawn for (2.5.5). I t should be noted again here, however, that the right-hand member of (2.1.15) is the slope at xi of the parabola through the points (xi-1 , ( x i , yj), and (xi+l, and that in consequence these slopes of parabolas represent a smoothing of the quantities mi. T h e fact that mi represents a smoothing of the quantities ( fj - f j - l ) / h is in marked contrast to the behavior of the Mi’s. It may be seen from (2.1.7) that the second divided differences f[xjPl , xj , xj+,] themselves represent a smoothing of the quantities M j . It is important to take these characteristics into account in applications of the spline to problems in which smooth second derivatives are required. One such class of problems involves the determination of flow patterns of a compressible gas by the “streamline” procedure. Here it is necessary to obtain streamline curvatures that are smooth as well as accurate, for the stability of the numerical procedure is usually involved. T h e spline has proved to be an effective tool in determining second derivatives by the following device: first determine streamline slopes by the usual method; then spline-fit the slopes themselves and use the resulting derivatives as the second derivatives required. Example 2.5.1 employs a typical streamline for a convergent-divergent nozzle to illustrate the effect of this device. Here M j represents the usual spline second derivative; y; represents the spline-on-spline second derivative. I n both cases, end conditions M , = M , , MNPl= MN have been used. T h e integral of the spline over {a, b} results directly from (2.1.2). We obtain the relation sA(x)

dx =

fi-1 + f i 2

hi - Mi-1

+ Mi hj3 .

24

(2.5.6)

Thus we are led to the formula

For the case of unequal intervals it is preferable, as indicated in the case of numerical differentiation, to evaluate the moments M iby the spline algorithm in Section 2.1 and in this way evaluate the integral on the left.

2.5.

45

APPROXIMATE DIFFERENTIATION AND INTEGRATION

When the intervals [ x + ~ ,xj] are of equal length, Eq. (2.5.7) becomes

For the periodic spline, this is simply the trapezoidal rule, since MI M , f ' * * M N = 0. If we sum the left-hand members of (2.1.5), we obtain for the nonperiodic spline the equation

+

+

fi-fo.

(2.5.8)

hl

Thus, for equal intervals, (2.5.7) becomes s,(X)

dx

=h

(Afo

+

+ + + M I + Mw-1 + 2'N)a

%fi + f 2

-(h3/72)(2Mo

*'*

+fN-2

S f N - 1

+A f N ) (2.5.9)

For the end conditions in which A0 = p N = a , which are those of principal concern to us, we have A;: = A-'N - i , N - - j . Set E3 3 E,(a)

= 2Ai.t

+ AT,: +

+ 2A;l3 .

Then (2.4.11) gives the equation (2MO

+ +

MN-l

+ 2MN) + (6/hz)(2f0 ==

EO[dO

+ + (6/hz)(f0

+fl

dN

+fN-1

+ 2fN)

+fN-afl-afN-l)l N

- (18/h2)

E3f3

'

1=0

By (2.4.6), we obtain the relation EO = [2PN(.)

-$QN-~(S)

+ (-2)1-NQ1(a) + (-2)2-NaI//

which, by (2.4.14), is shown to yield Eo =

[(2

+ a).

(2

+

-uN(2 f f u - l ) ] ( l - uN)(2 f f u y- 0 2 N ( 2 au-1)Z

+

+

+

5) -

2

(1

-

A I,

+

+ + u N ( 2 + a+) 0)

ffu

Similarly, we find for 0 < j < N that (2.5.10)

*

46

11.

We note further that E,

=

+ { 2 - [l

THE CUBIC SPLINE

1 - E,/4. Thus, (2.5.9) becomes

- (cu/12)1E01h(f1 + f N - l )

+

N-2 j=2

(l

+

kE?')fj

'

(2.5.11) A rapid procedure for calculating the numerical values of these coefficients for arbitrary a: is afforded by the following algorithm. Set G, = (-2)"D, so that Gn+,

Also set Fzn =

2Gn-2 3Gn-1

+ 4Gn+1 + Gn

+ Gn-3 + Gn-2 '

F2nt1

=

= 0.

5Gn-2 5Gn-,

+ Gn--3

+ Gn-2

I t may be verified directly that

and that, for 0

<j < N ,

Thus,

We include a table of these quantities for small n, Table 2.5.1. Coefficients for the case do = dN = 0, a: = 0 are given in Table 2.5.2 and those for do = dN = 0, a: = -2 in Table 2.5.3. T h e elements in the first of these tables were given previously by Holladay [1957] and are presented here again for convenience. Numerators of coefficients are displayed in the body of each table; denominators are indicated at the left of each row. Curiously enough, this gives the trapezoidal rule, Simpson's rule, and the three-eighth's rule for one, two, and three intervals. I n Chapters V and VI there is a detailed examination of the fundamental relationship between splines and approximate integration formulas, as well as the approximation of continuous linear functionals in general.

2.5.

47

APPROXIMATE DIFFERENTIATION AND INTEGRATION

TABLE 2.5.1

-3 -2 -1 0 1 2 3 4

-

4 -1 0 1 -4 15 -56 209 -780 2,911 - 10,864 40,545 - 151,316 564,719

5

6 7 8 9 10

2 -1 2 -7 26 -97 362 -1,351 5,042 -18,817 70,226 -262,087 978,122

-1 -1 5 -19 71 -265 989 -3,691 13,775 -51,409 191,861 -716,035 2,672,279

TABLE 2.5.2 4 1 1 i 3 1 0 3 & 4 11 11 4 z'p 11 32 26 32 11

__ 56

41

15 43 37 37 43 15 118 100 106 100 118 41

161 137 143 143 137 161 56

& 153 440 374 392 386 392 374 440 153 & 209 601 511 535 529 529 535 511 601 209 TABLE 2.5.3 ~~~~~~~~~

i l l $ 1 4 1 i 3 9 9 3 13 44 30 44 13 9', 35 115 90 90 115 35 2, 16 53 40 46 40 53 16

,as

3-go 131 433 330 366 366 330 433 131 179 592 450 504 486 504 450 592 179 163 539 410 458 446 446 458 410 539 163

4:~

-2 1

- 41 -5

-z 5 -_ 19 -1 26

15 -_ 71

6 -927

7 1 -265

_ _39672

48

11.

THE CUBIC SPLINE

I n Example 2.5.2, we indicate the application of the spline approximation to the evaluation of first and second derivatives of the function sin x on the interval [0, 2x1. Interval lengths of 20” and 10” are used. Second derivatives using the M i and also spline-on-spline second derivatives are indicated. T h e use of a variety of end conditions permits the estimation of the extent of the effect of end conditions. We consider four cases: (a) Mo = M , = 0; (b) Mo = M I , = M,; (c) yh and y k prescribed at their correct values; and (d) spline periodic. Because of symmetry, only half of the entries are exhibited. T h e spline-on-spline calculation for y” is exhibited for cases b and d in which the same type spline is employed as in the basic fit. T h e improvement in y” so calculated is evident by comparison with the sin x entries. T h e quality of the periodic y‘ and periodic spline-on-spline y” for both 10” and 20” intervals is noteworthy.* EXAMPLE 2.5.2 FITTING OF SIN x, 10” INTERVALS, CASESa, b ~-

M a = MN

=

Mo=M,, M N - ~ = M N

0

Case b

..

18x/r

sin x

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

0.00000 0.17365 0.34202 0.50000 0.64279 0.76604 0.86603 0.93969 0.98481 1.ooooo 0.98481 0.93969 0.86603 0.76604 0.64279 0.50000 0.34202 0.17365 0.00000

Y’

Y”

1.00005 -0.00000 0.98478 -0.17494 0.93966 -0.34212 0.86606 -0.50119 0.76603 -0.64516 0.64279 -0.76706 0.49999 -0.86924 0.34201 -0.941 07 0.17366 -0.98814 0.00000 -1.00184 -0.17367 -0.98825 -0.34199 -0.94061 -0.50000 -0.86997 -0.6428 I -0.76654 -0.76603 -0.64551 -0.86602 -0.50030 -0.93966 -0.34348 -0.98482 -0.17408 - 1.00003 -0.00023

y” (spline-on-spline)

Y‘

y”

1.00700 0.98292 0.9401 5 0.86593 0.76606 0.64278 0.49999 0.34201 0.17366 0.00000 -0.17367 -0.34199 -0.50000 -0.6428 I -0.76603 -0.86602 -0.93966 -0.98482 - 1.00003

-0.13797 -0.13797 -0.35203 -0.49854 -0.64587 -0.76687 -0.86929 -0.94 106 -0.98814 -1.00184 -0.98825 -0.94061 -0.86997 -0.76654 -0.6455 1 -0.50030 -0.34348 -0.17408 0.00023 -

-0.09541 -0.18053 -0.33132 -0.50507 -0.64101 -0.76661 -0.86588 -0.93968 -0.98471 -1.00018 -0.98467 -0.93954 -0.86628 -0.76594 -0.64272 -0.49993 -0.34202 -0.17396 +0.00018

* T h e excellent quality of the periodic spline-on-spline second derivative was obtained using 6 decimal places in the data and the approximation, valid for large N,

+

Y:’ = C, (9/4h2)[d,”D(z 2/3”2)/3”211[ft+~+e - 2f+3

+ f2+9-21.

2.5.

APPROXIMATE DIFFERENTIATION A N D INTEGRATION

49

10” Intervals, Cases c, d -

Periodic Y‘

Y”

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

1.ooooo 0.98479 0.93965 0.86606 0.76603 0.64279 0.49999 0.34201 0.17366 0.00000 -0.17367 -0.34199 -0.50000 -0.64281 -0.76603 -0.86602 -0.93966 -0.98482 -1.00003

-0.00093 -0.1751 9 -0.34205 -0.50121 -0.64516 -0.76706 -0.86924 -0.94107 -0.988 14 -1.00184 -0.98825 0.94061 -0.86997 -0.76654 -0.64551 -0.50030 -0.34348 -0.17408 -0.00023

Periodic Y” (spline-on-spline)

Y”

1.OW05 0.98478 0.93965 0.86606 0.76603 0.64279 0.49999 0.34201 0.17366 0.00000 -0.17367 -0.34199 -0.50000 -0.6428 1 -0.76603 -0.86602 -0.93966 -0.98482 - 1.00003

0.00000

-0.oooo0

-0.17494

-0.17375 -0.34196 -0.50002 -0.64280 -0.76600 -0.86594 -0.93970 -0.98479 -0.99996 -0.98479 -0.93970 -0.86594 -0.76600 -0.64280 -0.50002 -0.34196 -0.17375 -0.00000

-0.3421 2 -0.5001 9 -0.645 16 -0.76706 -0.86924 -0.94107 -0.98814 - 1.00184 -0.98825 -0.9406 1 -0.86997 -0.76654 -0.6455 1 -0.50030 -0.34348 -0.17408 -0.00023

20” Intervals, Cases a, b -

. -

Mo = MN = 0 18 x / n

sin x

0 2 4 6 8 10 12 14 16 18

0.00000 0.34202 0.64279 0.86603 0.98481 0.98481 0.86603 0.64279 0.34202 0.0oooo

Y’

Y“

0.99989 -0.0 0.93962 -0.34533 0.76600 -0.64950 0.49995 -0.8748 1 0.17363 -0.99489 -0.17362 -0.99472 -0.49996 -0.87505 -0.76599 -0.64916 -0.93962 0.34569 -0.99992 0.00022

M o= M I M N - = ~ MN Y‘

y”

1.02734 -0.27235 0.93227 -0.27235 0.76797 -0.66905 0.49942 -0.86957 0.17377 -0.99629 -0.99435 -0.17366 -0.87515 -0.49995 -0.649 17 -0.76599 -0.34569 -0.93962 -0.99992 0.00022

+

Case b y”(sp1ine-on-spline)

-0.19056 -0.35391 -0.62260 -0.87575 -0.98078 -0.98612 -0.8653 1 -0.64285 -0.34200 -0.0001 1

50

11.

THE CUBIC S P L I N E

20" Intervals, Cases c, d Periodic

Periodic

~

18x171

Y'

0 2 4 6 8 10 12 14 16 18

1.ooooo 0.93959 0.76600 0.49995 0.17363 -0.17362 -0.49996 -0.76599 -0.93962 -0.99992

-_____

Y"

Y'

-0.00105 -0.34504 -0.64957 -0.87479 -0.99489 -0.99472 -0.87505 -0.64916 -0.34569 -0.00022

0.99989 0.93962 0.76600 0.49995 0.17363 -0.17362 -0.49996 -0.76599 -0.93962 -0.99992

__-

Y" 0.00000 0.34533 0.64950 0.87481 -0.99489 -0.99472 -0.87505 -0.64916 -0.34569 -0.00022

y" (spline-on-spline)

-

0.00000 -0.34198 -0.64279 -0.86589 -0.98469 -0.98469 -0.86589 -0.64279 -0.34198 0.00000

2.6. Curve Fitting T h e specific objective of much of the development of one-dimensional splines is, of course, the fitting of a curve. Nevertheless, curve-fitting remains an art, and it is necessary to point out some of the techniques and artifices employed on splines used in the practice of this art. I t is generally desirable to employ more or less uniform distributions of mesh points when this is practical. When a long and short interval are in juxtaposition, oscillations frequently result which are not attributable directly to the data. T h e effect of the end condition chosen for a given arc, when this choice is within one's discretion, dampens rapidly as one moves in from the extremities of the arc. T h e necessity to determine more or less accurately the slope at the end of the arc, however, is a problem one frequently meets in curve fitting. T h e choice of end condition does have some effect upon the value of this slope. If flex points appear to be appropriate at the extremities, use M,, = M , = 0. I n the absence of other motivation, our own choice is Mo = M I , MN-l = M,. T h e resulting fit, however, may exhibit a strong behavior on the part of the quantities M j near the extremities with which this particular end condition appears to conflict. Frequently, in this situation, the more general end conditions M o = AoMl, M , = P,M,-~ may be employed with A,, and p N adjusted so as to be consistent with that behavior. A related problem concerns the fitting of an arc near an end of which the slope and curvature are both numerically decreasing, e.g., y = x1/2, 0 x 10, near the end x = 10. This behavior does not represent a natural runout for a cubic, and the fit will not be good if the mesh

< <

2.6.

CURVE FITTING

51

distance increases materially as we approach this end. Likewise, a good fit by a spline at the x = 0 end of this arc is prevented by the existence of a vertical tangent there. Some of the difficulties arising in the curve-fitting may be eliminated by a suitable change of coordinates. Many problems are intrinsic, however, and a technique of application of the spline is required which circumvents the difficulties presented. A highly effective procedure is the use of a parametric representation. Suppose Pi (xi, yi)( j = 0, I, ..., N ) represent N 1 points appearing on an arc C in the order given. If si is the cumulative chordal distance,

+

i=l

we spline-fit x versus s and y versus s. If desired, we can then approximate numerically the lengths of the segments of the resulting curve x = x(s), y = y(s) and construct a new fit of x and y against the cumulative arc length, although this step usually results in no perceptible change in the curve itself. A rather striking example is the fitting of x and y against cumulative chord length for the unit circle using periodic splines. Using eight points (45" apart) gives a maximum error of 0.00112 in the radius. For 12 points (30" apart), the maximum error in the radius is 0.000165. As a matter of interest, even for 4 points the error is less than 1 yo. We remark that with this device we have effectively accomplished an intrinsic parametric spline representation. T h e advantages in such a representation are many. T h e geometric configuration of the arc is of much less concern than for the conventional spline. Fitting a simple closed curve by a periodic spline in polar coordinates Y and 8 with respect to some pole inside the curve may be effective if no half-line from the pole intersects the curve more than once or is tangent to the curve. T h e arc-length representation presents no such limitation. We may fit curves as well in three dimensions, forming splines for x , y , and x in terms of chord length or arc length. Care must be taken, of course, that the curves being fitted actually possess the continuity conditions required. Using a single spline curve to fit an arc consisting of a piece of a circle and a tangent line at its extremity results in oscillations near the junctions due to the curvature discontinuity. Slope and curvature discontinuities (in two dimensions) when the fit is y versus x should be handled by terminating the splines at these locations or by appropriate spacing of points near the point of curvature discontinuity. T h e latter itself should not be a mesh point. I n the use of a parametric fit employing cumulative chord length, the

11.

52

THE CUBIC SPLINE

situation is far less critical. Appropriate spacing suffices even though the curvature discontinuity location is taken as a mesh point.

2.7. Approximate Solution of Differential Equations T h e spline may frequently be used to advantage in the solution of initial- and boundary-value problems in ordinary differential equations. T h e method to be described is of general application, but we restrict our discussion here to a two-point boundary-value problem for a secondorder equation. Consider the problem

+ p(x)y’ + q(x)y a,y(a) + a,y‘(a) a,,

Ly

= Y”

= ).(

(a

< x < b),

(2.7.1) + b,Y’(b) = bo Let us introduce here the cardinal splines associated with boundary condition (2.1.123). T h e cardinal splines are a set of N + 3 independent =

b,Y(b)

*

splines forming a basis for all cubic splines on the mesh A : a = xo < x1 < < x N = b. We define these in the following way: A d , k ( (k ~ )= 0, 1,..., N ) and B,,,(x) (k = 0, N ) are cubic splines on d with A ~ , k ( X ~ ) = 8 k , ~ ( j =1,o..., , N ) , Ai,k(X,)=O(i=OandN), Bd,k(x3) =O(j

= 0,

1,..., N ) ,

k

B;,k(xz)= S k , z ( i= 0 and N ) , k

=o, 1,..., N , =0

(2.7.2) and N.

Here 6,,j is the Kronecker delta. We may express the spline satisfying end condition (2.1.18i) and interpolating on d to the solution of the differential equation in the form SA(Y; 4 =

c N

AA,?(X)Y(X3)

3=0

+ Y’(aPA,o(x)+ Y ’ ( b ) B d , N ( X ) .

(2.7.3)

Set EA(y;x) equal to the difference between y ( x ) and S A ( y ;x): (2.7.4) q y ; x) = Y ( 4 - SAY; 4. If there exists a solution y ( x ) of (2.7.1) which is of class C2[a,b] and is

unique, then E y ) ( y ;x)

=o

(11 d \Iz--“)

( a = 0, 1, 2) uniformly with respect

to x in [a, b]. If f ( x ) possesses a higher order of regularity, then the

approach of E y ) to zero is correspondingly higher in accordance with the results of Section 2.3. Using (2.7.4), we form the difference LS,(y; ). - r ( x )

= =

+

%(Y; x) p(x)S,’(y;). GAY; 4,

+ q(x)S,(y;x) - r(x)

2.7.

53

APPROXIMATE SOLUTION OF DIFFERENTIAL EQUATIONS

where GA(y;x) = --Ei(y; x) - p(x)EA’(y;x) - q(x)E,(y; x). We have G,(y;.x) = o(ll I 11). Substituting from (2.7.3) into (2.7.1) now gives the equation

+

N

+

G ~ ( yx). ;

(2-7.5)

Thus the ordinatesyj = y ( q ) ( j = 0, I , ..., N ) and the slopes yb y h = y‘(b) satisfy the equations

= y’(u),

y(xj)LAA,j(x)

i=O

N

C

yjLAA,j(xz)

y’(a)LBA,O(x)

= .(xi)

+ azyd

= an

biy,

+

=

-

+ y&BA,&xz)+ y h L B ~ , ~ ( x i )

J=O

aiyo

Y’(b)LBA,N(X)

f G,(J’; x i )

( i = O , 1 , 2 ,..., N ) , bzyi = bn

(2.7.6)

-

We introduce the notation

GA

(Y; ,yo 1 yi Y N y&IT, = (0, G,(y; ‘ ~ o ) , . . . G&; , xN),

RA

= (a0 T ( x g ) , . . . , y ( x N ) ,

YA

y-..,

1

bnIT,

and let H A represent the matrix a2

...

0

a1

LBA,o(xo) LAA,o(%) LAA,l(xo)

I

H A = [

0

’.. LAA.N(xo)

LBA,N(Xo)

0

1.

LBA.,(~M)L A A A ~ N )L A A . I ( ~ N )... LAA.N(xN) L B A , N ( ~ N ) ... 0 0 0 h, b2

Then we have HAYA= RA

+ GA.

T h e quantities G,(y; xi) are, of course, unknown. T h e method consists of replacing these by zero and then determining from HAY: = RA the approximation Y: of YA. If we have the property that H i 1 exists and that I/ H;lG, 11 + 0 as /I d I/ + 0, then the solutions of the modified equation HAYS = R , (2.7.7) define splines that converge uniformly to y(x) on [u, b]. I n particular, this conclusion follows if the norms I/ H;l 11 are uniformly bounded. For purposes of illustration, we now consider the problem y“ - azy

= 0,

0

y(0) =y(l)

< x < 1, =

1.

(2.7.8)

54

11.

THE CUBIC SPLINE

Here H Ais the matrix 0

0

1

0 - a2

Ad).N(xo)

''_

AI;,N(Xl)

1

0

0

Bi.N(xo) B i ,N(x1)

1.

... 4 , N ( x N )- a2 B ~ , ~ ( J c ~ )

&(xN)

0

_''

0 0

Premultiply the members of Eq. (2.7.7) in this case by the ( N 3 ) x ( N 3 ) matrix

+

+

(2.7.9) 0

0

0

"'

1

where A is the coefficient matrix in (2.1.8) with A, = p N = 1. T h e resulting equation is ... 0 1 0 0

I

6

_ _6

hi

0

hi2

6 hi(hi

I

+ hd

2012

- p1012

_6 _ -

-6

0

0

0

0

0

0

...

0

0

hZ(hZ

+ h3)

-

hz(hi

+ hz)

hzh3

0

0

0

0

0

0

0

0

0

0

-

Xlaz

6 - 2a2 --

p2"z

0

-6hNZ

6

2012

hihz

0

...

0

X,a2

hi2

... ...

2.7.

55

APPROXIMATE SOLUTION OF DIFFERENTIAL EQUATIONS

+

I n these equations, replace -yo’ (yl - yo)h, by u0 ,yN’- (yN- yN-J/hN by u N . Interchange the first and second equations of the resulting set, and the last and next to last. T h e modified equations are -h,a2

0

...

0

0

...

6

1;

~-

hih2

2a2

6 hdh,

+ hz)

0

...

0

0

...

0

- hlE2

0

0

..

..

1

0

... 0

Here the coefficient matrix is clearly diagonal dominant if 11 A 11 is sufficiently small, and the norm of the inverse matrix is uniformly bounded. I n this process, the error vector Gd is multiplied by the matrix (2.7.9), which has a bounded norm, and by the inverse of the coefficient matrix in (2.7.10), which likewise has a bounded norm. Thus, convergence of the splines to y ( x ) follows; indeed,

I/ [ ~ ’ ( x o -) Y ; * , Y ( X ~ ) -YO*,***,Y(~N) -YN*,Y’(~N> -YJ

11-0

as 11 A 11 -+ 0. Table 2.7.1 gives, to five decimal places, the exact solution of the x 1, y(0) = y(1) = 1, together with problem y” - lOOy = 0, 0 the spline approximations employing 10 and 20 intervals. T h e exact solution is y = [cosh 1O(x - O . ~ ) ] / C O5S. ~Because of the symmetry, only the first half of the interval [0, 11 is covered.

< <

TABLE 2.7.1 X

Exact y

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

1 .ooooo 0.60657 0.36799 0.22332 0.13566 0.08263 0.05070 0.03 170 0.02079 0.01520 0.01 348

y(N

=

10) y ( N

1.ooooo 0.35107 0.12343 -

0.04390 0.01075 -

0.01065

=

20)

1.ooooo

0.60333 0.36406 0.21975 0.1 3277 0.08042 0.04906 0.03049 0.01987 0.01444 0.01277

y’(N

=

20)

56

11.

THE CUBIC SPLINE

One should note also at this point that, once the spline solution has been completed, the information required for spline interpolation between mesh points is available. This is particularly significant when, as part of a longer calculation, the solution of the boundary-value problem is required at various locations in the interval [a, b ] . An important instance also is the use of an automatic plotter that frequently requires interpolation at a great many intermediate points. Nonlinear differential equations are effectively handled by the method of splines when used in conjunction with the quasilinearization techniques introduced by Bellman and Kalaba [1965]. Here the problem Y"

= f(x, y , Y ' ) ,

< x < b;

a

4 Y ( 4 Y'(41 4 0, BCY(b),Y'(b)l

=0

is solved by the iterative procedure Yn"+l

= fV@,

Yn YA)(Yn+1 - Y n ) +f,& !

Yn YXY?!L+l7

with the linearized boundary conditions 4CYn(O),Y?!L(O)I[Yn+1(0) - Yn(0)I B,[Yn(O), Y;(o)l[Yn+l(o) - Yn(0)I

+ 4 Y n ( O ) , r?!L(o)l~YA+l(o) - Y 3 9 l = 09 + W Y n ( O ) , Y?!L(O)I[YA+l(O) YA(0)l = 0. -

For the case of equal intervals, the moments for the cardinal splines introduced previously may be obtained directly from the results of Section 2.4. We have end condition (2.1.18i), SO that A, = p N = 1 in the coefficient matrix A of (2.1.8). Thus,

B6,0(x,)

A;*o(xo) A;,o(xl) A;.o(xz)

A;,l(xo) A;,l(x,) A;,lbd

d,0(XN-2)

A;;,O(XN-Z)

A;.l(XN--I)

B;,,(XN-l)

A;.o(xN-l)

A; ,I(XN-1) A;.l(xN)

-B;.o(xo) B;.O(xl)

A;,o(xN)

3;dXN)

--6/h 0 0 0

-

'.' .'.

". '. '

A;.N(Xo)

B;;.N(XO)

A;.N(xl)

&.N(Xl)

A;"%)

&.N(4

A;.N(XN-I)

&V(WI)

A;;,N(XN-I)

."

0

0

0 0 0

... ... ...

...

&. NbN-1) &.N(XN)

0 -6/h 6/h 0 0 3/h2 3/h2 -6/h2 3/h2 3/h2 --/ha 0 0 3/ha - 6/hZ 0

0

0 0

"'

-

-

... ... ...

- 6/h2 3/hz 3/h2 - 6/h2 3/h2 0 0 0

0 0 3/h2 0 -6/h2 3/h2 6 / h -6jh

0 0 0 6/

2.8.

APPROXIMATE SOLUTION OF INTEGRAL EQUATIONS

57

T h e elements of the inverse matrix A-l are given in Section 2.4. We have, in fact, + $-?) A-1 = (a-i + uZ)((J3--N (0 < i < j < N ) , 2,3 ( 2 + u)(u-N - u") AT1

-

z,N -

A-l OeO

=

(u-2

+ ui)

(2

+ u)(u-N

(2

+ u)(u-"

- U-N)

(0 < i


2

- 0")

with A:'a .J = A;,: (0 < i < N , 0 < j < N ) and A,; = A;?9,N-i for all i and j . These quantities can be evaluated using the methods set forth in Section 2.4 or directly coinputed from the property that uk u+ and upk

-

uk are solutions of the difference equation u,+~

+ + 4 ~ , ++~ u,

= 0.

2.8. Approximate Solution of Integral Equations Consider the linear integral equation

(2.8.1)

< <

< <

wheref(x) and k(x, t ) are continuous, a x b, a t b. Introduce 1 cardinal splines defined by the end conditions Mo = M I , the N M N p 1= M , , here designated A d , j ( ~j) ,= 0, 1,..., N . Thus,

+

T h e spline of interpolation of this type is

58

11.

THE CUBIC SPLINE

Here E,(x) = o(1l d t i u ) iff(.) and k ( x , t ) are of class Ca[a, b] (01 = 0, 1, 2, 3). We proceed much as in the case of the linear differential equation in the preceding section. We determine y o ,y1 ,..., y N by replacing G,(xj) by 0 for j = 0, 1,..., N . Thus,

T h e integrals

Ij,i= h Jb

a

q x j , t)A,,i(t)dt

are evaluated in closed form if this can be done conveniently [note that A, ,*(t) is piecewise cubic] or approximated numerically. If the latter course of action is followed, we note that, inasmuch as the cardinal splines A, ,Jt) have already been determined, little additional work is required to spline-fit each k ( x j , t ) for this integral approximation. I t is required then to solve the system of equations

T h e convergence of the splines so obtained as 11 A I/ -+ 0, in the case in which 1 is not a characteristic value of the homogeneous integral equation, rests on the corresponding proof for the case in which trapezoidal integration is used (cf. Goursat-Bergmann [1964, p. 46 ff]). From an algebraic point of view, it is usually simpler to employ the spline in its standard form (2.1.2) in making this application to integral equations. As an example, we consider the integral equation associated with the two-point boundary-value problem (2.7.8): (2.8.2)

where k ( x , t ) = 2 ( x - 1)t = LYqt -

1)

(0 < t

< x), (x < t < 1).

2.8.

59

APPROXIMATE SOLUTION OF INTEGRAL EQUATIONS

We replace y ( x ) by the spline Sd(x) determined by

f o r i = 0 , 1 , ..., N . We have, of course, y o = y N = 1, and we use this fact to reduce the number of equations by 2. For 0 < i < N we obtain yi

=

1

+

.“Xi

-

1)

J:;,-~

c /Mi-l i

j=1

+ a2xi 1 N

JQ1

Xi-1

j=i+l

(xi - t)3t dt 6hj

( x .- t)3 (t-1 ) 3 dt 6hj (t

9‘

- 1 ) ( ~ j-

hj

xj-1

X.

Xi-1

( t- 1 )

~ -dt ~ ) ~ t

6hj

(t 6hj

dt

t )dt

hi

Xj-1

Xi-1

+MjJ ’

( t - l)(t - xj-1) dtl

‘3

+ M ijx? ( t - ~

.

T h e integrals are evaluated, and we obtain directly for equal intervals (hi= h , j = 1, ..., N ; Nh = 1) y.

=

[

1

i-1

1 - a2(N - i) [h5 - - jMi 12 j-1

1 . 7 zMi- -- (Mo - Mi)] 24 360

--

j=1

-a%

lh5

[-

1

-

N-1

1

12 j=i+l

1 7 ( N - j ) M j - - ( N - i)M, - -(&“-Mi)] 24 360

N-i

N-l

By multiplying respective members of the Znd, 3rd, ..., ith equations of (2.1.8) by 1,2, ..., (i - 1) and adding, we obtain

c j Mj. - 3 1 [

i-1

i=1

- -

-

1

M,

+ 1 iMi-l 1 (i-l)Mi 1 + h-1 [yo- iriPl + (i - l)yi], -

60

11.

THE CUBIC SPLINE

and, similarly, N-1

1

1=2+1

(N -j)M

1

--

j-3

f

[1

1 -MN

2

3[ Y N

+ j1( N

-(N

-

1

1 i)Mt+l - Z(N- i - l ) M i

- i)Yt+l

+(N -

- I)Yi]*

With an application of (2.1.5), the term in Mi-,, Mi, Mi+1,namely ( N - i)i 72 Mi-l

N - i)i + ( ( N 18- i)i -0) + ( N 72 Mi,, Mi

1

becomes

We note that M o = a2y0= a2, M l = a2yN= a2,so that we are involved with end condition (2.1.18ii). T h e quantity M iis expressed in terms of y1 ,..., yN-, by means of (2.4.11) using the quantities A,:for end conditions (ii) (Section 2.4):

There results the following equation system for the quantitiesy, ,..., Y ~ - ~ when N is even: for i = 1, 2,..., N/2,

where

T h e comparison with the exact solution (see Section 2.7) is given in Table 2.8.1 for the case in which N = 20. I n the present instance, it is evident from (2.8.2) thaty"(x) - a2y(x)= 0. I n the general case, the end condition for the spline is determined by differentiation in the integral equation to obtain a restriction on Mo and M N .

2.9.

61

ADDITIONAL EXISTENCE AND CONVERGENCE THEOREMS

TABLE 2 8.1 Exact y

X

-

~-

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0 35 0.40 0.45 0.50

Spline integral equation method ~-

1.00000 0.60657 0.36799 0.22332 0.13566 0.08263 0.05070 0.03 170 0.02079 0.01520 0.01348

-

~~

1.00000 0.60656 0.36802 0.22334 0.13568 0.08264 0.05070 0 03170 0.02080 0.01 520 0.01348

T h e present example includes a considerable part of the structure employed in the general case in which the kernel k ( x , t ) is itself splinefitted at each level xi in order to approximate the integrals involved.

2.9. Additional Existence and Convergence Theorems The existence of the spline of interpolation was demonstrated in Section 2.2 under the assumption that 1 A, 1 < 2 and I p N I < 2 in (2.1.8) or under comparable restrictions in (2.1.16). I n order to extend these existence results, it is necessary to examine in some detail properties of the coefficient matrix in (2.1.8) or (2.1.16). We compiete here the proof of the general existence theorem.

Theorem 2.9.1. The periodic spline S d ( x ) on the mesh A : a = x,, < x1 < < x N = b with prescribed ordinatesy, , y1 ,..., y N = yo always exists. The nonperiodic spline with end conditions (2.1.18iii) [or (2.1.1.19iii)l existsprovidedh, < 4 and p N < 4 (or p, < 4 and AN < 4).

It remains to complete the proof €or the nonperiodic situation in which we require only that A, and p N be less than 4. (The condition po < 4, A, < 4 is equivalent to this one.) For k > j, we define

,

(2.9.1)

11.

62

THE CUBIC SPLINE

together with the conventions D(Aj ,..., A,) = 2, 1 , 0 for k = j , k = j - 1, k = j - 2, respectively. For k j + 1, we use the Laplace expansion for determinants and obtain the relation D(A,

)**.I

'k)

=

D(Aj

Thus, for k 3 j

-

9

A,+l)D(Aj+2

'k)

- 2A7+1(1

- h3+2)D(Aj+3

?...)

(2.9.2)

+ 1, we have in the notation of continued fractions 1 D(4 A,+,) 7

2h+lU - A,+,) 2-

&+A1

-

2-

A,+,>

... hc--l(l - h k ) 2

(2.9.3)

T h e relationship between tridiagonal matrices (continuant matrices) and continued fractions which is entering here is indicated by Aitken [1958, p. 1261. T h e left-hand member of (2.9.3) is a linear fractional function of each of its arguments on the hypercube 0 Xi 1 (j i k). Its denominator is different from zero on this hypercube by Gershgorin's = A, = 0, we have D(Aj ,..., A,) = 2k-j+1 theorem. When Xi = (k > j - 1). Thus, D(Aj,..., A,) is positive on the hypercube. It takes on its maximum and minimum values for the hypercube at points where each hi is 0 or 1. = 0, the left-hand member in (2.9.3) is equal to 1/(4 - hi). When is 0 or 1. Hence, if 0 Xi 1 When Aj+l = 1, it is or according as ( j i ,< k), we obtain the inequality

< <

<

fr

< <

< <

(2.9.4)

If we note that

we obtain by induction the following lemma.

2.9.

63

ADDITIONAL EXISTENCE AND CONVERGENCE THEOREMS

Consider now the coefficient determinant in (2.1.8),D = D (Ao ,..., A N ) , expanded first using minors of the first row and then using minors of the last row:

We find

D

= (4

-

A0)(4 - A,)D(A,

,...,AN-2)

,..., A N - 2 , O ) = 2(4 - AN)D(I, A 2 ,..., AN-2) = 4D(1, A, ,..., AN-2, 0)

= 2(4

- Ao)D(A2

From (2.9.5),it now follows for A. determinant D in (2.1.8) satisfies

for A, = 0 and AN-l

=

1,

and AN..,

= 0,

for A, = 1

and AN-,

=

1

and AN-,

= 0.

for A, for A,

=0

=

1,

< 4 and p N < 4 that the coefficient

T h e proof of the theorem is now complete. We turn to the extension of Theorems 2.3.2 and 2.3.3 to cover more general end conditions. For this purpose, it is desirable to use properties of the elements of the inverse matrix which are of considerable interest for their own sake. These properties generally relate to the rate of decay of the magnitudes of the elements of the inverse matrix as we move away from the principal diagonal. T h e elements of the inverse A-l of the coefficient matrix in (2.1.8) are obtained directly from the cofactors of A. Thus, A,:

= (-l)i+9(Ao

)...)Ai-,)AiAi+,

... Aj-,D(Aj+, ,...)A,)/D

(0 < i <j
(2.9.8) A,:

=

(-I)i+'D(A,,,..., Aj-l)(l-Aj+l) ... (1 -Ai)D(Ai+,,..., A,)/D (O<j
where A, means 1 - p N , and again D denotes the coefficient determinant. We require two lemmas on the properties of these elements. If we expand D(Aj ,..., Ak) for j < k in minors of the first row, we obtain D(Aj ,..., A k )

= 2D(Aj+,

,..., A,)

- ( I - Aj)xj+,D(hj+2)...,A k ) ,

64

11.

THE CUBIC SPLINE

so that

provided D(A, ,..., Ak) # 0. This ratio is equal to $ when Aj+l = 1 and to 2/(4 - A j ) when Aj+l = 0. T h e ratio is a bilinear function of each of 1 forj
< <

If we expand D(h, ,..., hk) in minors of the last row, we obtain, for 0 < hi 1 ( j < i < k) and hi < 4,pk < 4, the inequality

<

0

If we examine the proof of (2.9.4) we see immediately when

< hi < 1 ( j < i < k) and Aj < 4, pk < 4 that we have

A similar derivation yields, under the same conditions, the inequality

T h e first of the lemmas required is as follows.

< hi < 1 (0 < i < N ) , p N < 4 , and

Lemma 2.9.2. For A, < 4, 0 any integer p , 0 < p < N , we have

2.9.

where p,

Proof.

ADDITIONAL EXISTENCE A N D CONVERGENCE THEOREMS

=

I

For

-

65

(2.9.14)

A,.

p odd,

By (2.9.9) and (2.9.1 I), this ratio lies between the two quantities min

[A2’1-4

2 -

A,

.

1

(P-1)/2

,

max

[A,2

1-4-A.2

.

(P-l)/2

.

For p even,

which, by (2.9.1 I), lies between the quantities

This completes the demonstration of (2.9.13). Inequa!ity (2.9.14) is similarly obtained using (2.9.10) and (2.9.12). It should be noted that inequalities (2.9.13) and (2.9.14) are sharp. For suitable values of the quantities h i , the equality actually obtains. 0

Lemma 2.9.3. For A,

< p < q < N , we have

< 4, 0 < Ai < 1 (0 < i < N ) , p N < 4, and

66

11.

THE CUBIC SPLINE

which may be written as

< <

<

if 0 p q - 2 and q N , since we define D(A, ,..., A,) = 0 if j = i - 2. We obtain for the second member of (2.9.15), when 0 < p < q - 2 and q < N , the value

{W,+l

,***,

Ag-l)lF1

{D(X,+l

,*.*>

A-1)

{D(Av+l,..., A,)-l

-

,..., A,-z)>-’

&A,-,D(A,+,

for A,

- &(l- A,+l)D(A,+z ,..., A,-l)}-l

{D(&+i ,.*.)Api)

-&(I

-

for A, = 0 and A, = 1,

-

=0

and A, = 0,

for AD = 1 and A,

=

1,

iAplD(&+i ,..-,Ag-2)

A,+1)W,+z

I...)

Ag-1)

+ ax,-,c1

- A,+l)W,+2

for A,

, - * * > A-2)1-l

=

1 and A,

= 0.

We have used here the properties D(0, A,, ,...,A,) = 2D(A,+l ,...,Ak), D(Ai ,..., Ak-,, 1 ) = 2D(A, ,..., A,-,). A routine examination of the cases A,, = 0 or 1, AqPl = 0 or 1 , and the application of (2.9.9) and (2.9.10) yields

Use of Lemma 2.9.1 gives (2.9.15). We turn now to the extensions of Theorems 2.3.2 and 2.3.3 to more general end conditions.

Theorem 2.9.2, Let f ( x ) be of class C’[a, 61. Let {Ak) be a sequence of meshes on [a, b] with linzk+a:11 A , / / = 0. Let sdk(x) be a spline of interpolation to f ( x ) on A , satisfying end conditions (2.1.19iii) with inf,(4 - pk,0) > 0, infk(4 - A,,) > 0, and p k , o and bounded as

k-+ 00. (a) If

2.9. as k

-+

03,

67

ADDITIONAL EXISTENCE AND CONVERGENCE THEOREMS

then SLk(x)converges uniformly t o f ‘ ( x ) on [ a , b], and

[s$’(x) -f”’(x)]

=

1 -’)

o(I1

( p = 0 , 1)

(2.9.18)

uniformly with respect to x in [ a , b ] . (b) If c , , ~ and c , , ~ , are bounded US k -+ 03, then (2.9.18)is valid on any closed subinterval of a < x < b, and {Sdk(x)) + f ( x ) uniformZy on [a, b].

Proof (a). We introduce the ( N ,

+

+ 1) x ( N , + 1) matrix G, , ... ... ...

(1 - ~ k , d / 9 0

0

3

0

0

L

...

... ...

0

3

0 3 0 (1 - X k . N k ) / 9

0 0 0

0

0

+.

and rewrite (2.1.16) in the form (2.9.19) , where I , is the unit matrix of order N , 1, and B, is the coefficient matrix in (2.1.16) associated with A , . T h e right-hand member of (2.9.19) is equal to the vector Bkmk

-BkGkCk = (Ik -BkG,)ck

+

(2.9.20 )

Now c k , i / 3 (0 < i < N ) is equal tof’(5) at some 5 in xi--l < x < xi+l. Thus, the norm of the vector (2.9.20) does not exceed the larger of the two quantities 1

3 1 ER) I

$-

+pro 2P(f’; / /

11)

+ 3 P ( f ’ ; 11

II),

68

11.

THE CUBIC SPLINE

From (2.9.8) and Lemma 2.9.3, we see that 11 B,’ I/ is bounded with respect to k. Thus, it follows from (2.9.19) that, as k + 00,

I/ mk 7Gkck 11 Also,

I/ G k c ,

1

+

3 ck,O

f

- [f’(Xk,o),f’(Xk,i),‘..,f’(Xk,Ny)l - Pk.0

9

ck,l - f ’ ( a )

= -1E L

3

+

O*

II

+

0, since

- Pk*O

3

[F

-fya)] ,

T h e remainder of the proof now follows precisely the pattern of proof of Theorem 2.3.2. ~ assumed Proof (b). Suppose now that the quantities c , , ~ and c , , ~are merely to be bounded as k + co and that p k , o and hk,Nksatisfy this condition also. For an interval [a’, b’] with a < a’ < b’ < b, there are at least n; = 1 + [(a’ - a)//ld, 111 mesh points in A , at or to the left of a’, and nl = 1 + [(b - b’)/ll A , 111 at or to the right of b’. I n forming the sums representing the components of mk - G,c, from (2.9.19) associated with mesh locations within [a’, b’], it is seen that the first, second, next-to-last, and last components of (2.9.20) are multiplied, respectively, by quantities not exceeding

[See (2.9.8) and Lemma 2.9.2.1 Inasmuch as n; and n; increase indefinitely as k -+ CO, we obtain

1 m k , j - f ’ ( ~ ~ , I}~ = ) 0. lim { a ’ $max xk,j
k+m

T h e uniform convergence of (Sdk(x))and (Si,(x)> to f(x) andf’(x) on [a’,b’] now follows in the standard way.

2.9.

ADDITIONAL EXISTENCE AND CONVERGENCE THEOREMS

69

I n order to demonstrate uniform convergence of {Sdk(x)}to f ( x ) on [a, b], we remark that the method of proof for (a) suffices to show that the quantities m k j are bounded with respect to k under the conditions of (b). T h e uniform convergence of {SA,(x)}to f(x) on [a, b] is now a direct consequence of (2.1.10). We note that the term containing and Y , , ~is a cubic with extrema at xk,j-l and x k , j and hence monotone between. T h e validity of the following corollaries is also evident.

Corollary 2.9.2.1. Under the conditions of Theorem 2.9.2, if f ’(x) satisJies a Holder condition on [a, b] of order OL (0 < 01 l), and if = O(1l A , ] l a ) , then the right-hand member o f (2.9.18) = O(I1 d, is repZaced by O(l1 A , ill+a-p) ( p = 0, 1).

<

Corollary 2.9.2.2. I f f ’ ( x ) is continuous on [a’, b’], a < a’ < b’ < b, and exists and is bounded f o r all x in [a, b], then {SL,(x)}-+ f ’(x) uniformly on [a’, b’], and {Sdk(x)}+f(x) uniformly on [a, b]. T h e proof of the next theorem follows very closely the pattern of proof employed for Theorem 2.9.2 and will not be repeated.

Theorem 2.9.3. Let f(x) be of class C2[a,b]. Let {A,} be a sequence of meshes on [a, b] with limk+m11 A , 11 = 0. Let SA,(X)be a spline of interpolation to f(x) on A,, satisfying end condztions (2.1.18iii) with infk(4 - Ak,o) > 0, infk(4 - ~ L , , ~>J 0, and and p k , N ,bounded as k+m. (a) If c; = dk,,, -

(2

+ &,o)f”(a)-O

and

6;

= dk,Nr

-

(2

+

plc,NJfb(b)+O

as k + co, then Sik(x)converges un;formly to f “(x) on [a, b ] , and we have

I s p )-f‘”’(.) I

=

.(I1 A , 11””)

(P

= 0,

192)

(2.9.21)

uniformly with respect to x in [a, b]. (b) If dk,o and dk,Nkare bounded as k -+ co, then (2.9.21) is valid on any closed subinterval of a < x < 6. Moreover, {S;,(x)} and {SAk(x)} converge uniformly t o f ’ ( x ) a n d f ( x ) on [a, b]. T h e analogs of the two corollaries carry over as well.

Corollary 2.9.3.1. Under the conditions of Theorem2.9.3, i f f ”(x) satisfies a Holder condition on [a, b] of order a (0 < 01 1 , andif 6; = O(ll d, 1P)

<

70

11.

THE CUBIC SPLINE

and c: = O(/lA , Ip), then the right-hand member of (2.9.21) is replaced by O(Il A , Il2+.-”) ( p = 0, 1,2).

Corollary 2,9.3.2. I f f “(x) is continuous on [a‘, b’] ( a < a‘ < b’ < b) and exists and is bounded f o r all x in [a, b], then {Si,(x)}+f”(x) uniformly +f (PI(.) uniformly on [a, b] f o r p = 0, 1. on [a’, b’], and {SL:)(x)) Other convergence results related to these can clearly be obtained. Existence off “(x) can be replaced by uniform boundedness of difference quotents. I t is possible to obtain convergence at a single point when f”(x) is known to exist at that point (Ahlberg and Nilson [1963]). We turn now to a consideration of third and fourth derivatives.

Theorem 2.9.4. Let f(x) be of class C3[a,b]. Let {A,) be a sequence of meshes on [a, b] with

p II A , ll = 0 -tm

and

”P[Il A , ll/mp(x,,j

- x,,9-1)1

=B

<

Let the splines of interpolation {Sdk(x)) to f(x) on A , satisfy end condition (2.1.18iii) with infk(4- Xk,J > 0, infk(4 - F , , ~ , ) > 0, and and F k , N k bounded as k + co. If

+ (2 +

- [(4- pk,N,)’?€,N,-I

I*.k.Nk)hk,Nk-llf”’(b),

both approach 0 as k + 03, then on [a, b] we have

[szyX)- p y x ) ]

= .(II

A,

(p

113-q

I f f ”(x) satisfies a Holder condition of order are both O(ll A , lib), then on [a, b]

01

= 0, I, 2,3).

(0 < a

< 1) and if

(p

i,2,3).

[sspyx) -f(qX)] = o(llA , 113+a-9

= 0,

E;

and

ei

Proof. T h e first and last of Eqs. (2.3.21) are here replaced by the general forms (dropping mesh index k): [6 - (2

+

+ (2 +

~O)P1ICL1~1

X0)hl29

=

(2

+ h)4- 3d0 hl + h,

,

2.9.

71

ADDITIONAL EXISTENCE AND CONVERGENCE THEOREMS

If the resulting coefficient matrix is decomposed as was the matrix C of (2.3.21), we obtain again E D - F , where D is now 6-(2+A0)p1

(2+&)& 1

PI

+ + Pl

[

...

0 0

0

!0

... ."

0 A,

A2

...

PN-2

0

0 0

0 0

1

+ PN-2 + AN-I + P N ) PN-1

AN-i

(2

-

(2

+! PNLN)AN-,

1.

This matrix has dominant main diagonal and a uniformly bounded inverse if infk(4 - A k , o ) > O and infk(4 - p k , N , ) > 0. Thus, 11 C-l 11 is uniformly bounded. We fotm the Nk x N k matrix 1 1

H=

-

+ ( 2 + A,)

[(4 - A,)

0

1 0

0

0 0

[o

0

A,] 0

0

0 0 0

"'

...

1

...

...

0

... 0

1 1 - [(4

-PN)~N-I

Then, if r represents the vector

we have C(u - Hr )

and the right-hand vector is

= (I -

CH)Y,

0' 0 0

+ (2 + PN)

0 PN-11

1.

72

11.

THE CUBIC SPLINE

T h e rate at which li(I - C H ) r 11 approaches zero is now evident, and the conclusions of the theorem follow from the boundedness of 11 C-l I/. Our final result concerns a rather curious property of convergence to the fi”(x). For the sake of simplicity, we restrict our attention to the periodic cubic spline, although the argument can be carried through for nonperiodic splines as well. We consider the jump in the spline third derivatives at xj and set

From the set of equations for the quantities ai ,

we obtain, by subtracting corresponding members of the j t h from the ( j 1)th equation and dividing the result by hj + hi+l+ the equation

+

+

DS

=g,

(2.9.22)

2.9.

ADDITIONAL EXISTENCE AND CONVERGENCE THEOREMS

73

and the columns by

We obtain as a result the matrix

<

If we assume for the meshes d under consideration that11 d ll/minjl hi 1 /3 < 00, we find again that 11 D-l 11 is bounded with respect to k . T h e sum of the elements in th ejth row of D is

+

+ h+l)+ + PjPj+l

G , = P32_1Pj2P3+lej L’jPjPj+l(& 3

’j-l’j

’j-lXjZX+1ej+l

If the meshes A become asymptotically uniform as k max hj 3

-

4I

---f

+ co, that

is, if

0,

then maxi/ Gj - 8 I -+ 0. I n general, let G be the diagonal matrix with Gi the diagonal element in the j t h row. Then for G-lD - I, the sum of elements in each row is zero. Write Eq. (2.9.22) as D6

-

G-lDg

= (I - G-lD)g.

-

Letfiv(x) be continuous on [a, b]. T h e quantities 4gj differ from fi”(xj) by an amount O(li d 11). Thus, /!(I - G-’D)g + 0 as 11 d 11 0. Since 11 D-’ /I is uniformly bounded, it is evident that 11 6 - D-lG-lDg 11 -+ 0.

74

11.

THE CUBIC SPLINE

If the meshes become asymptotically uniform as 11 A I/ -+0, then I/ 6 - 2g 11 + 0. Thus we have proved the following.

Theorem 2.9.5. L e t f i " ( x )be continuous on [a, b ] . L e t {A,} be a sequence of meshes on [a, b] with limk-m I/ A , /I = 0 and sup,ll A , l\/[miniI h,,i I] =

p < co. If, in

addition, the meshes become asymptotically uniform as

k + co, we have

CHAPTER

rrr

Intrinsic Properties of Cubic Splines

3.1, The Minimum Norm Property T h e treatment of cubic splines in Chapter I1 does not reveal the intrinsic structure of spline functions. Historically, too, this structure was well hidden; more than a decade elapsed after the introduction of the spline by Schoenberg [1946] before the first of the intrinsic properties was uncovered. This property, which we refer to as the minimum norm property, was obtained by Holladay [1957]. Before proceeding with the statement and proof of Holladay’s theorem, we require some notation and terminology. By S n ( a , b), we mean the class of all functions f(x) defined on [a, b] which possess an absolutely continuous (n - 1)th derivative on [a, b] and whose nth derivative is in L2(a,b). We denote by Sp“(a, b ) the subclass of functions in S n ( a , b) which, together with their first n - 1 derivatives, have continuous periodic extensions to (- 03, co) of period b - a. A functionf(x) is of type I’ if it has a first derivative that vanishes at x = a and x = b. Two functions are in the same type I equivalence class if their difference is of type 1’.A spline function S(x) can be represented in many ways in terms of a finite number of parameters. If two of these parameters are S’(a) and S’(b),we say the representation is of type I and that S(x) is of type I when it is represented in this manner. I n a similar fashion, we say a functionf(x) is of type 11’ if it has a second derivative that vanishes at x = a and x = b. Two functions are in the same type 11 equivalence class if their difference is of type 11’. A type I1 representation of a spline S(x) is a representation into which S”(a) and S”(b) enter explicitly. We say that S(x) is of type I1 when it is represented in this manner. T h e purpose of these definitions of “type” is to facilitate the discussion of cubic splines; they are modified later when splines other than cubic splines are considered. T h e minimum norm property is expressed in Theorem 3.1.1, which follows, in a form slightly stronger than originally stated. T h e proof, however, is that of Holladay. 75

76

111.

INTRINSIC PROPERTIES OF CUBIC SPLINES

Theorem 3.1+1. Let A: a = x, < x1 < < x N = b and = ( y i 1 i = 0, I , ..., N } be given. Then of all functions f ( x ) in Z 2 ( a ,b ) such that f ( x i ) = y i , the type 11’ cubic spline SA(Y ; x) minimizes Y

Jt

If “(XI l2

dx.

Moreover, SA(Y ;x) is the unique admissible function that minimizes this integral.

Proof. If f ( x ) belongs to S 2 ( a ,b), and iff(xi)

jIi f ” ( x ) =

-

= yi

, then

S J Y ; x) I z dx

j b I f ” ( x ) l 2 dx

-

a

2 / b f r ’ ( x ). S ; ( Y ; x) dx a

+

b

a

I S i ( Y ;x ) l2

j I f ” ( x ) l a dx-2 j b{f”(x)-Si(Y;x)} - S i ( Y ;x) d x - l b b

=

a

a

a

dx

1 S;(Y; x) l2

dx.

We have, however, Jl{f”(x) - S i ( Y ;x)} . S:(Y; x) dx

since Sy(Y;x) is constant on each mesh interval [ x i - - l , xi] and f ( x i ) = Sd(Y ;x i ) = y i (i = 0, 1,..., N ) . In virtue of the continuity of { f ’ ( x ) - S i ( Y ;x)) S,“(Y;x) on [ a , b ] , it follows that

3.2.

THE BEST APPROXIMATION

77

PROPERTY

T h e last equality is true in view of the hypothesis that S,(Y; x) is a type 11’ spline. I t now follows that

I S i ( Y ;x) l a

dx

=

I f “ ( x ) - S i ( Y ;x)

l2

dx.

(3.1.1)

T h e right hand member is positive unless Si(Y ; x) =f”(x) a.e., i.e., unless S,(Y; x) -f(x) Ax B , which reduces to f(x) since S A ( Y ;xi) = f ( x i ) ( i = 0, N ) . This proves the theorem. Equation (3.1.1) is called the jirst integral relation. T h e content of Holladay’s theorem was anticipated to a degree by researchers in the theory of elastic beams dating back to the Bernoullis and Euler, (Sokolnikoff [1956, p. l]), but the abstract formulation, the simplicity of Holladay’s proof, and the integral relation (3.1.1) that this method of proof establishes represent a major contribution. Holladay, however, did not pursue the subject further and did not explore the far-reaching consequences of the first integral relation.

+

+

3.2. The Best Approximation Property Let us introduce the pseudo-norm (3.2.1)

/If-

into -X2(a, b) and, for fixed f(x) in X 2 ( a , b), consider SAll,where SA(x)is a cubic spline of prescribed type with respect to a fixed mesh A : a = xo < x1 < ... < xN = b. T h e question arises, “Does the spline S , ( f ; x) of interpolation to f(x) on d minimize Ilf - 5, /I ?” We have given an affirmative answer to this question for several important situations in Section 2.3; we formulate this result for the periodic case.

Theorem 3,2.1, Let A : a = x,,< x1 < *.. < xN = b and f(x) in X:(a, b ) be given. Then of all periodic cubic splines s A ( x ) , s A ( f ;X) minimizes I l f - S, 11. If S,(x) also minimizes I l f - S, I/, then Sd(x) = S A ( f;X) const.

+

T h e proof of Theorem 3.2.1 contained in Section 2.3 is classical in nature and requires the determination of stationary points. I n Section 3.4, we generalize Holladay’s argument and obtain an elegant proof that avoids determining stationary points and extends beyond the periodic case not only to type I splines and type I1 splines, but to situations where the splines involved are not simple. Although we are not yet in

78

111.

INTRINSIC PROPERTIES OF CUBIC SPLINES

position to define a simple spline in complete generality, we say a cubic spline is simple if it is in C 2 [ a ,b]. T h e property expressed in Theorem 3.2.1 is called the best approximationproperty (Walsh et al. [1962]). Like the minimum norm property and a number of other important intrinsic properties that we subsequently develop, it can be obtained as a simple consequence of the first integral relation. This integral relation itself (as is apparent from Holladay’s proof of the minimum norm property) is a consequence of a very general identity involving spline functions, and results when certain restrictions are imposed upon the splines involved. T h e identity plays a fundamental role in spline theory and allows it to proceed smoothly in situations where methods such as the standard minimization argument employed in the proof of Theorem 3.2.1 become very cumbersome. However, when direct methods, such as those employed in Chapter 11, are applicable, they generally yield sharper results. This is particularly true with respect to rates of convergence. We now obtain the indicated identity explicitly and use it as a cornerstone for a theory of splines.

3.3. The Fundamental Identity We can obtain the identity just mentioned and at the same time make more transparent the conditions under which the first integral relation is valid, if we again transform (as in Holladay’s argument) the integral

Jl

{ D ” f x ) - D2S,(x)}D2S,(x) dx

by integrating by parts twice. T h e operator notation D”f(x) rather than f”(x) will make the generalizations in Chapter VI more natural. T h e xi result of this double integration by parts for each interval xi-l x of the mesh A : a = xo < x1 < < x,,, = b is

< <

3.4.

79

THE FIRST INTEGRAL RELATION

since D4SA(x) is identically zero on each open mesh interval xiPl < x < xi. Consequently,

in view of the continuity of (Of(.)- DS,(x)} D2S,(x) on [a, b]. Generally, however, D3SA(x)is not continuous at the mesh points of A ; this is the reason for the presence of the summation

Substituting (3.3.1) in the identity

llf

- g /I2 = Ilfl12 - 2

with g(x)

= Sd(x)

W(4- D 2 g WW

X ) dx -

Ii g 1i2

13-34

gives the fundamental identity

which is valid for anyf(x) in Z 2 ( a ,b) and any simple spline S,(x) an arbitrary mesh A : a = xo < x1 < < x N = b.

on

3.4. The First Integral Relation When SA(x)= S,(f;x)

+ const, the summation

c{f@) N

i-1

-

s,(4)D3S,(4IX*

Xt-1

in the fundamental identity vanishes. If in additionf(x) and SA(x)are in Z : ( a , b), or f(x) - SA(x)is of type I’, or S,(x) is of type 11’, then the fundamental identity reduces to the first integral relation. These conditions are clearly only sufficient and not necessary, since, in particular, end conditions of mixed type may be imposed. We can

80

111.

INTRINSIC PROPERTIES OF CUBIC SPLINES

even relax the continuity requirement on Si(x) and still obtain the first integral relation by requiring Sd’(x) to interpolate to f ’(x) at the mesh points of A ; even this does not exhaust the possibilities. T h e following theorem is, in view of these remarks, a direct consequence of the fundamental identity.

Theorem 3.4.1, If f(x) is in 3tr2(a,b ) and S A ( f ;x) is a spline of interpolation to f(x) on a mesh A : a = xo < x1 < < xN = b and any of the conditions, (a) f ( x ) and SA(f; x) are periodic, (b) f(x) SA(f; x) is of type 1’, (c) Sd(f;x) is of type IT, is satisjied, then ~

I1 S ~ , S l l 2+ llf - S A , f \ I 2 . I n the light of Theorem 3.4. I, let us re-examine Theorems 3.1.1 and llfl12

=

3.2.1. Since in Theorem 3.1.1 we have S,(Y; x) 3 S A ( f ;x) for any f(x) in X 2 ( a ,b) such that f ( x i ) = yi (i = 0, 1,..., N ) when S,( Y ;x) and S,,(f; x) are both of type 11’, Theorem 3.4.1 implies

iif!i2

- //

SA.Y//2 =

I/.#-

sA,Ylla

>, O.

Consequently, we have

jl I S i ( Y ;x) l2 dx < s:

l f ” ( x ) I 2 dx

for any f ( x ) in %?(a, 6 ) such that f ( x i ) = yi (i = 0, 1,..., N ) , with equality if and only if S i ( f ;x) =f”(x) a.e. Thus, the minimum norm property is a direct consequence of the first integral relation. As we have already seen Holladay’s argument constitutes a proof of this relation. If y o = y N and both f(x) and S,(x) are restricted to ,Xp2(a, b), then Theorem 3.4.1 implies the following.

Theorem 3+4.2. Let A : a = xo < x1 < < x N = b and Y = {yi 1 i = 0, 1,..., N ; y o = y N } be given. Then of all functions f(x) in %;(a, b) such that f(xi) = yi (i = 0, 1,..., N ) , the periodic spline SA(Y ; x) minimizes

J; If’W I2dx

and is the unique admissible function that minimizes the integral. I t should be observed that, ify, = y N , if SA(Y ;x ) is~the periodic spline of interpolation to Y , and if Sn(Y ; x) is the corresponding type 11’ spline of interpolation, then

3.4.

81

T H E FIRST INTEGRAL RELATION

where we have equality only when

since S,,(Y; x ) and ~ S,(Y; x) interpolate to the same values at x = a and x = b. I n this sense, Theorem 3.1.1 is stronger than Theorem 3.4.2. We can formulate yet another analog of Theorem 3.1.1.

Theorem 3+4.3, Let d: a = xo < x1 < ... < xN = b and Y = = 0, I , ..., N } begiven. Then of allfunctionsf(x) in ,X2(a,b) such that f ( x i ) = yi (i = 0, 1,..., N ) and f ’ ( x i ) = y ; (i = 0, N ) , the type I spline S,( Y ;x) minimizes {yb, y;V, yi 1 i

Moreover, S,( Y ; x) is the unique admissible function that minimizes this integral.

Proof. All but the uniqueness follows directly from Theorem 3.4.1. T o see the uniqueness, observe that, if g(x) also satisfies the condition of Theorem 3.4.3 and minimizes (3.4.1), then Theorem 3.4.1 implies that IIg - S,,Yl12

=

Ilg1l2- I1 S A , Y l 1 2 = 0

+

or g ( x ) = Sd(Y ;x) + Ax B = S,( Y; x), since g(m)(a)= SFJ(Y ;a ) for 01 = 0, 1. If the requirement that Si(Y ;xi) = y ; (i = 0, 1) is omitted, the type 11’ spline of interpolation to Y on d minimizes (3.4.1). A spline function S,(x) on d of a given type depends in a linear fashion on its values at the mesh points of d and on the values of its derivatives at x = a and x = b. This is evident from (2.1.2), (2.1.8), and (2.1.9). ConsequentIy, we have, under a variety of conditions,

although the decomposition is not unique. Some useful sets of end conditions which serve to make the decomposition (3.4.2) both valid and unique are as follows:

+

SAf; 4 - f ( x ) , S,(g; x) - g ( 4 , and SA(f g ; x) -f(4 are all of type 1’. (b) SA(f ;x), S,(g; x), and S,( f g ; x) are all of type 11’. g ; x) are all periodic. ( c ) SA(f;x), SA(g ;x), and Sd(f

(a)

+ +

-

g(4

82

111.

INTRINSIC PROPERTIES OF CUBIC SPLINES

I n particular, we have S,(f - s,;). = S,(f; ). - S ,(.)

(3.4.3)

for a number of end conditions on SA(f - S,; x) and S,( f;x). We can choose S A ( f - S,; x) to be periodic, of type 11', or such that S,(f - S,; x) - f(x) SA(x)is of type 1', and not only have (3.4.3) valid but also have the fundamental identity in each case reduce to

+

1 .f

- sA

11'

- 11

SA,f-sA

11'

=

l1.f - sA

- Sd,f-s,j

11'

=

Ilf

= llf

- sA -

-sA,f

S,,f

+

sA

11'

11'7

which implies that l i f S, /I 3 i l f S A , f11. When S,(f- S,; x) is of type 11', we can modify condition (b) for the decomposition (3.4.2) and only require that S,(f; x) - SA(x) be of type 11'. This establishes Theorem 3.4.4, which is a generalization of Theorem 3.2.1. Observe that the strongest result is obtained whenf(x) - S,(f; x) is of type 1', since in this case no restriction is placed on S,(x).

Theorem 3.4.4. Let d: a = xo < x1 < ... < xN = b and f(x) in Z 2 ( a ,b ) be given. If S,(x) and S , ( f ; x) are splines on d such that one of the conditions, (a) f(x), S,(x), and S,(f;x) are in Zp2(a, b), (b) S,(x) - S,(f; x) is of type 11', (c)f(x) - S A ( f x) ; is of type I', is

satisfied, then

llf

-

s, II 2 Ilf

- SA,f Il.

If we have equality, then S,(x) = S,( f;x) + Ax periodic case, where A

= 0.

+B

except in the

T h e decomposition (3.4.2) results from the possibility of finding a set of parameters upon which a spline S,(x) depends linearly and which, together with continuity requirements, serves to define SA(x).We refer to these parameters as defining values.

3.5. Uniqueness I n the present chapter, we have been relying hitherto on the results of Chapter I1 for the uniqueness and existence of the various splines under discussion. T h e assertions of all our theorems are correct and do not depend for their correctness on either existence or uniqueness, although, if we did not have at least existence, the theorems would be vacuous. A number of existence and uniqueness theorems of

3.5.

83

UNIQUENESS

Chapter I1 were obtained with relative ease due to the dominance of the main diagonal in the matrices involved. Indeed some of the important boundedness properties needed for convergence theorems of Chapter I1 were also obtained from this dominance. T h e investigation of splines of higher odd degree contained in Chapter I V is severely hampered by the absence of this diagonal dominance, and only limited existence and uniqueness theorems are obtained. I n Chapter V, these difficulties are circumvented through an application of the first integral relation (in more general form). I n order to see this very useful application of the first integral relation, we proceed to establish the basic uniqueness and existence theorems for cubic splines by this method of argument.

Theorem 3.5.1. Let A : a = xo < x1 < < xN = b and f ( x ) be given. If any of the conditions, (a) f ( x ) and SA(f;x) are periodic, (b) f ( x ) - Sd(f;x) is of type I', (c)f(x) - S A ( f ;x) is of type IT,is satisjed, then Sd(f;x ) is unique. Proof. Suppose that Sd(f;x) and Sd(f;x) are two splines of interpolation to f(x) on A , both of which satisfy the same one of the conditions a, b, or c. I n each of these cases, Sd(f;x) - S A ( f ;x) is a spline of interpolation to the zero function Z(x) such that the first integral relation holds, in virtue of Theorem 3.4.1. Thus,

I/

- sA,f f

11 s A , f

=

sA,f

- sA,f

11'

//

= -

1 ' - I/ s A , f /I sA,f

- sA,f

- sA,f

11'9

1.'

I t follows that S,(f; x)

2

S,(f;x)

+ Ax + B.

Since both Sd(f;x) and.SA(f; x) are splines of interpolation tof(x) on A ,

A=B=O.

I n the proof of Theorem 3.5.1, the function f(x) served only to , determine an interpolation vector Y to which Sd(f;x) and S A ( f x) interpolate on A . T h e first integral relation was not applied tof(x) but to Z(x); consequently, the differentiability or even continuity of f ( x ) is immaterial, andf(x) may be regarded as an arbitrary function on [a, b].

REMARK3.5.1. T h e uniqueness asserted in Theorems 3.1.1, 3.4.2, and 3.4.3 implies the uniqueness of the splines there considered. Condition c, however, requires the slightly more elaborate argument given here.

84

111.

INTRINSIC PROPERTIES OF CUBIC SPLINES

3.6. Existence We are now in a position to give an alternative proof of the existence of type I, type 11, and periodic splines of interpolation which carries over to more general situations.

< xN = b and f(x) be Theorem 3.6.1. Let A : a = xo < x1 < given. Then there exist type I and type 11 splines Sd(f ; x) of interpolation to f(x) on A in each type I and type 11 equivalence class. If f ( a ) = f ( b ) , there exists a periodic spline of interpolation to f(x) on A . Proof. Since we consider only the values of f(x) at mesh points, we can modify f(x) so that its values on A are unchanged but the modified function f(x) is in any specified type I or type I1 equivalence class. From Theorem 3.5.1, we can conclude that, if S4(kx) exists, it is unique. T h e splines in question will exist if the matrices in (2.1.8) and (2.1.9) are nonsingular. Both of these equations are of the form A.M=Y

(3.6.1)

where A is a matrix and both M and Y are vectors. T h e components of M are the values of Si(f;x) at the mesh points of A . If (3.6.1) had two distinct solutions M , we would have two distinct splines of interpolation to f(x) on A-both periodic, both in the same type I equivalence class, or both in the same type I1 equivalence class. This, however, would contradict Theorem 3.5.1; consequently, (3.6.1) has a unique solution M . Since a unique solution of (3.6.1) for one Y implies the uniqueness of the solution for any Y , A-l exists, and this proves the theorem. Although Theorem 3.6.1 establishes the existence of A-l, the method of proof does not place a bound on 11 A-l/j. I n this sense, Theorem 3.6.1 is inferior to the existence theorems of Chapter 11. Its superiority lies in its generality. Moreover, the first integral relation together with the second integral relation, which will be introduced in Section 3.9, to a very large degree compensates for the lack of the bound on I/ A-l 11.

3.7. General Equations T h e matrix equations (2.1.8) and (2.1.9) obtained in Chapter I1 were derived from the hypothesis that Si(x) is continuous and piecewise linear. I n Chapter IV, similar equations are obtained for higher-order splines of odd degree; but only under restrictions such as uniform spacing do these equations assume a simple form. There is, however, in

3.7.

85

GENERAL EQUATIONS

all cases a system of approximately N equations in the same number of unknowns, where N is the number of mesh intervals. T h e representation of higher-order splines of odd degree in terms of these quantities is considerably more complex than in the case of cubic splines(cf. Section 4. I). T h e analogous procedure for generalized splines, which are investigated in Chapter VI, is unclear, since the linearity assumption is not valid. Consequently, each case requires special analysis. Even in the case of generalized splines, it is possible to write down in a straightforward manner a system of 2nN equations in 2nN unknowns where again N is the number of mesh intervals and n is the order of the pertinent differential operator. Moreover, in each mesh interval the spline is easily represented in terms of these quantities. T h e matrices that arise have the property that only a limited number of subarrays contain nonzero entries. This allows the use of special inversion procedures on a computer which greatly reduce the storage required for performing the inversion. T h e matrices have the disadvantage, however, that, as the length of the mesh intervals approaches zero, the matrices approach a singular matrix. Moreover, when a special representation is possible, the system of equations is usually significantly smaller in size. I n order to derive these equations, let us alter our point of view and regard the construction of the cubic spline of interpolation as the problem of piecing together N solutions of the differential equation

Dy

(3.7.1)

= 0,

each solution valid in a different open mesh interval of the mesh A : a = x,, < x1 < < x N = b, such that the resultant spline Sd(f;x) interpolates to a prescribed functionf(x) on d, satisfies a definite set of end conditions, and is in ,X3(a, b). Any solution of (3.7.1) is a linear combination of four linearly independent solutions ui(x) (i = 1, 2, 3,4) that can be chosen such that @(O) = 8t-l,j ( j = 0, I, 2,3). Specifically, we have ui(x) = xi-l/(i - I)!. I t follows that, in any mesh interval xiPl x xi,

< <

(3.7.2) Since Sd(f;x i ) = f ( x i ) = yi , we have cil = yi-l (i = 1,2,3 ,..., N ) . For Sd(f; x) to be in ,X3(a,b), it is necessary that 4

1

j-1

C i j U Y ) ( X i - xi--l)

c 4

=

Ci+l,jU:k)(0) = Ci+l,k+l

j=1

(K

= 0,

1 , 2; i

=

1 , 2 ,..., N

-

1).

(3.7.3)

86

111.

INTRINSIC PROPERTIES OF CUBIC SPLINES

If S,( f;x) is of type I, we have (3.7.4)

4

C

3=1

4

cNjuj(xN

- xN-1) ==Y N

1cN~u;(xN

,

- xN-1)

j=l

= y;

-

(3.7.5)

2).

(3.7.6)

I n the periodic case, these equations are replaced by 4

cll = y o = y N ,

C c ~ ~ u : ~-) x( x~ ~- ~=)

c ~ , ~ + (k ~

= 0, I ,

j=1

Let us set

B=

[

1

-1

0

0 0 -

0

0 0

0 - 1 0

0 0

-1

0

0-

(3.7.7.2)

3.8.

CONVERGENCE OF LOWER-ORDER

The n

DERIVATIVES

AC = Y

87 (3.7.7.9)

where, for type I splines,

A=

.c:

0

A,

B

0

A,

0 0

0 0 B

...

0 0

0 0

and, for type I1 splines,

c:'

0 B

Al

A=[;

". ... ...

0 0 0

0 0 0 (a =

...

AN-^

0

0 0

"'

...

0

'..

0

B

O),

(3.7.7.10)

8=1),

(3.7.7.11)

=

B

4 # # i-l:]

0 0

0

"'

1,

c: 0

(a=O,

c;

and, for periodic splines, Al B 0 0 A, B

...

...

0 0

0 0

(3.7.7.12)

A =

with (3.7.7.8)replaced by =

(rl

9

0, 0, o,y, 0, * " 7

"',YN 1 0 ,0, O)=*

(3.7.7.13)

I n each case, the matrix A is nonsingular, since the representation (3.7.2) uniquely determines the coefficients c i i . Thus, as in the proof of Theorem 3.6.1, two solutions of Eqs. (3.7.7) would give rise to two distinct splines of interpolation; this would contradict Theorem 3.5.1. Since we have cil = yiP1 (i = 1, 2, ..., N ) , the size of the system of equations easily can be reduced from 4nN equations to 3nN equations.

3.8. Convergence of Lower-Order Derivatives We now investigate the convergence of a sequence of splines of interpolation { S d N , f(} N = 1, 2, ...) defined on a sequence of meshes

88

111.

INTRINSIC PROPERTIES OF CUBIC SPLINES

( A , : a = xo" < xl" < * * - < xm", = b) such that / / A , // 4 0 as N + 00. As in the convergence proofs of Chapter 11, it is not required that dNlC A N 2for Nl < N , . One important notion in the discussions that follow is that, since SA,( f ; x) interpolates to f ( x ) o n d , , we know from Rolle's theorem that in every mesh interval xpl < x <xiN(i = 1, 2,..., m,) there exists an xi, such that f'fXiN>

= s&(f;

(3.8.1)

Later we need the analogous result that in each interval

(i = 2, 3,..., m,) there exists an,3iN such that f"&,)

Theorem 3.8.1. {A, :a

xp2 < x < xiN (3.8.2)

= S&(j; * i N ) .

Let

= xoN

< x,N <

.**

<XZN

= b}

(N

= 1, 2,...)

and f ( x ) in Z 2 ( a , b) be given with /I A , /I -+ 0 as N -+ m. Let {SdN,f} ( N = 1, 2,...) be the corresponding sequence of splines of interpolation determined by one of the following conditions: (a) f ( x ) is in ,Xp2(a, b), and SdN( f;x) is periodic ( N = 1 , 2, ...), (b) f ( x ) - SdN(f;x) is of type I' ( N = 1, 2,...), (c) SAN( f ; x) is of type II' ( N = 1, 2,...). Then we have Zim / f ' " ' ( x ) - ~F$(fi x) I

N-tm

=

o

(a = 0,

(3.8.3)

1)

uniformb for x in [a, b]. Proof.

Let xi, in [xp1 , xi"] be such thatf'(xi,)

=

Si,( f;x i N ) .T h e n

by Schwarz's inequality applied tof"(x) - SiN(f;x) and 1, Consequently, lf'(x) - s i N ( f i x) I

d llf

-

S A N , , II

*

I x - xi,

< K j x - xi,

Ill2

1 12,

(3.8.4)

where K = 211 f 11, by Minkowski's inequality and the minimum norm property. If A , C A , ( N = 1, 2,...), we may take K = I/f - SAl,!ll, since in this case SAl(f;x) is a spline on A N , and the best approximation

3.9.

89

THE SECOND INTEGRAL RELATION

property applies. Since there is an xiN in every mesh interval, we can find one such that 1 x - x i N I II dNll, which implies that Si (f;x) -+ f ’(x) as N 00 uniformly for x in [a, b]. Similarly for XF-; such that xEl x x i N ,

<

< <

-+

I j(x) - sdN(j; I

<j y I f ’ ( x ) - siN(j;x) I I dx I < +KII

113/2,

where 2 is equal to xEl or xiN, depending on which is closer to x. T h is proves the theorem.

Corollary 3.8.1.1. If d l C d N ( N = 1 , 2,...), IIdNII-+O as N-+co, and SAN( f;x) ( N = 1, 2,. ..) is in a prescribed type 11 equivalence class, then Zim StA(f; x) = f ( = ) ( x >

( a = 0, 1)

N-m

uniformly for x in [a, b ] . Proof. We may take K as imation property.

/If

-

Sdl,f in virtue of the best approx-

Corollary 3.8.1.2. Let Ax), ( A N } ( N = 1, 2,...), and ( S d N ( f x)) ; ( N = 1, 2, ...) satisfy the conditions of Theorem 3.8.1 or Corollary 3.8.1.1. Then

+ o(llA ,

fyx) = syyj;). N

p

2

9

( a = 0, 1)

(3.8.5)

uniformly for x in [a, b].

3.9. The Second Integral Relation Let us examine in more detail the pseudo-norm (3.9.1)

c jZ’{D”f(x) N

=

-

j=1

Xj-i

D”d(x)}z dx

(3.9.2)

90

111.

INTRINSIC PROPERTIES OF CUBIC SPLINES

for any spline SA(x) defined on a mesh A : a = x,,< x1 Moreover,

<

< xN = b.

Substitution of (3.9.3) in (3.9.2) together with the continuity of (Of(.) - DSA(x))(D2f(x)- D2SA(x))on [a, b] and the equation

shows that

(3.9.4)

T h e identity (3.9.4) is important in spline theory and is valid for any simple spline SA(x)on any mesh d: a = xo < x1 < *.. < xN = b and any functionf(x) in X 4 ( a , b). Under a number of conditions on Sd(x), this identity reduces to

which we refer to as the second integral relation. I n particular, the following theorem is immediate from (3.9.4).

< x N = b and f(x) in Theorem 3.9.1. Let A : a = xo < x1 < .X4(a,b) be given. If S,( f;x) is the spline of interpolation to f ( x ) on d satisfying one of the conditions (a) f(x) - S,(f; x) is of type 1’;(b) f(x) - SA(f;x) is of type II’; (c) S,( f;x) is periodic and f(x) belongs to .X:(a, b); then

3.10.

91

RAISING THE ORDER OF CONVERGENCE

REMARK3.9.1. If f ( x ) is in . X 2 ( ab), the statement ‘ t f ( x ) - S A ( f ;X) is of type 11”’ is not well defined, sincef”(x) may not exist at x = a or x = b. I n the present case, f”(x) is in S 4 ( a ,b), and so the statement is meaningful. 3.10, Raising the Order of Convergence

( N = 1, 2,...) satisfy the Let f ( x ) , ( A N ) ( N = 1, 2,...), and conditions of Theorem 3.9.1. By the argument in Section 3.8, we have \ f ( a ) ( x) S y i ( f ;x)

1

< ilf - SdN,f11 - 11 ~l,\l‘~-~”’/~

I),

( a = 0,

(3.10.1)

and, since the second integral relation is valid, (3.10.1) becomes

Setting 01 = 0 and substituting the resulting inequality in the right-hand member of (3.10.2), we obtain

If‘W - S:;(,(f; 4 I <

ir

I W x ) I dxlJ’

- Lp. sup If(.)

- SA,(f; x> IP * II 4,

b]

where

-’

Ji=i++, J

-

c,

J3

= {3(1

+ 8)

-

I J%

2a}/2.

(3.10.3.1) (3.10.3.2)

Again setting 01 = 0 in (3.10.3.1), substituting in the right-hand member of (3.10.2), and repeating this process, we obtain (3.10.3.1), but with

J,

=

H k a + ... + (Vk

J z = (2) J3 = {3(1

1

(3.10.4)

201u)/2 + & + + (if”-’) -

at the end of k steps. But, since this is true for any positive integer k, it follows that I ~ ( W

- s~;(,(f;x) I

G

j

b Q

I~

( x I dx )

. / IA ,

1i3--a

( a = 0,

1).

(3.10.5)

92

111.

INTRINSIC PROPERTIES OF CUBIC SPLINES

This result can be obtained in a more direct manner, for by (3.10.2) we have

If

there is nothing to prove; otherwise,

which establishes (3.10.5) for 01 = 0. Substituting (3.10.6) in (3.10.2) gives the general case. We have established by this argument the following theorem.

Theorem 3.10.1.

Let

{ A N : a = x N < xlN

<

***

< X "N N

= b}

(N

=

1, 2,...)

and f ( x ) in X 4 ( a , b) be given with I/ A , I[ -+ 0 as N - + 00. If S A N ( fx;) is a spline of interpolation to f ( x ) on A , and one of the conditions, (a) f ( x ) - SAN(f ; x) is of type I' ( N = 1, 2, ...), (b) f ( x ) - S d N ( f x; ) is of type 11' ( N = 1, 2, ...), (c) f ( x ) is in X j ( a , b) and SdN(f; x ) is periodic ( N = 1, 2,...), is satisfied, then

REMARK3.10.1. If we replace (3.10.1) by the tighter inequality -

Syd(f; I < ($)'-" /if. 11 A ,

'A,;,

11(3-h)/2

/I ( a = 0, I),

which was established in Section 3.8, then we obtain

(3.10.7)

3.1 1.

93

CONVERGENCE OF HIGHER-ORDER DERIVATIVES

3.1 1. Convergence of Higher-Order Derivatives Let f ( x ) in X 4 ( a ,b), d: a = xo < x1 < ... < x N = 6, and S,(f; x) be given. Then, by (3.7.2) for xiPl x xi (i = 1, 2,..., N ) ,

< <

S,(f; x) =

c 4

CijUj(X

- xi-1).

(3.1 1.l)

j=1

Let 6;[c, d] denote the ath equally spaced difference quotient of an arbitrary function g(x) on an arbitrary interval [c, d ] , and interpret 6,O[c, d] as g(c). Then there exists an xia in [xi--l , xi] such that

G&--l , xi1 = s;0(f;X i m )

(a = 0,

1,2, 3).

I t follows that

and when I] d Ij -+ 0 these equations approach*

Consequently, for I/ d 11 sufficiently small, the system of equations (3.11.2) is solvable, and each cii is a linear combination of the four quantities 8>A,,[~i-1,xi] ( a = 0, 1, 2, 3). We can, in addition, employ Theorem 3.10.1 to obtain the relations

I

, Xi] - S,”[.i-I

I

, xi1 I

(3.1 1.4.1)

= 0,

I

(3.1 1.4.2)

* Observe that the functions u J “ ) ( x ) ( j = 1 , 2, 3, 4; OL = 0, 1, 2, 3) are continuous on [0, b - a] and, hence, uniformly continuous and bounded on [O, b - a].As I1 A I1 -+ 0, these functions remain fixed and only the points at which they are evaluated vary.

94

111.

INTRINSIC PROPERTIES OF CUBIC SPLINES

a

We observe that an extra factor of can be obtained in the right-hand member of (3.11.5) in view of Remark 3.10.1.

Theorem 3.1 1.l. Let a sequence { A , :a

N

= xo

< XIN <

*-*

of meshes

<X i N

= b}

(N

=

1, 2,...)

with 11 A N I/ -+ 0 as N-+ co and f(x) in X 4 ( a ,b) be given. Let RAN= maxi=,,...,m,~~A N lI/(x," - xEl) be bounded with respect to N , and let (S4"(f;x)} ( N = I , 2,...) be a sequence of splines of interpolation to f ( x ) satzsfying one of the conditions, (a) f(x) - SAN( f ; x) is of type I' ( N = 1, 2,...), (b) f(x) - SAN( f ; x) is of type 11' ( N = 1, 2 ,...) and d, C d, , (c) f ( x ) is in -Xp4(a,b), and each S A N ( fx) ; is periodic; then f ( a ) ( x )= S2A(f;x)

+ O ( !A! ,

1 3-=)

( a = 0, 1, 2, 3).

(3.11.6)

un;formZy for x in [a, b]. Proof.

For some xz in S?[Xi_,

[xp,, xiN],we know

,Xi] = f y x : )

( a = 0,

1, 2, 3).

(3.11.7)

3.12.

95

LIMITS ON THE ORDER OF CONVERGENCE

Consequently, (3.11.4) implies

I G A N , , l l X i - - l , Xi1 I


I

w(X)

*

I

> Xi1

+ I axxi-,

1 dX 11 AN *

113--a

9

xi1 I

+

SUP xe[a.bl

I~(U’(.)

I

( a = 0, 1 , 2 , 3 ) ,

which, by the hypotheses of our theorem, is bounded as N+ 03. Since the matrix in Eqs. (3.11.2) approaches the identity matrix as N -+ co uniformly both with respect to the number and location of the mesh intervals [ x i - - l , x i ] , it follows that all the coefficients determining the splines S d N ( f x) ; ( N = 1, 2, ...) are uniformly bounded with respect to N . Therefore, 1 Si:(f; x)i is uniformly bounded for a < 4 and all N ; thus, there exists a real number B such that 1 Si:(f; )I. B for every N . Since in each mesh interval [xL2, xr] (i = 2, 3, ..., mN) there is an xiN for which SiN(f;x i N ) =f”(xiN),

<

I

f’w- siN(f;I <

Ij(3yX)

Jx X.

- s y y ; X)

ZN

< 2 {xzgPbl 1

+w

I

~

3

I I dx I

11 AN~ 11.

*

More generally, we have

if(-aw

-

s y ; ~ x;) I G 2 {xz;pb,

1

~

3

I

+ BI w

11 A ,

113--a

( a = 0,1,2).

I n addition, i f ( 3 ~ 4

- sy;(f; I G

1 f ( 3 w

I

+ BI

*

II ~ N I I O ,

which concludes the proof.

REMARK3.11.1. Observe that, for 01

=0

and 01 = 1, Theorem 3.10.1 gives more precise bounds on the rate of convergence. Perhaps even more important, no restriction is placed upon the meshes involved; in Theorem 3.11.1, RANmust be bounded as a function of N . 3+12. Limits on the Order of Convergence

In Sections 3.10 and 3.11, we established that f ‘ ” ’ ( ~= ) S yN) ( f ;X)

+ O(1l AN /I3-=)

(a = 0,

1, 2, 3),

(3.12.1)

96

111.

INTRINSIC PROPERTIES OF CUBIC SPLINES

and in Chapter I1 we established, by more special methods, that

although the mesh restrictions and constants involved are not the same in both cases. I n particular, the constant in (3.12.1) is proportional to V a b [ f ( 3 )whereas ], the constant in (3.12.2) is {whenf(x) is in C4[a, b ] ) proportional to llf(4) Jim; here we have used the standard notations

Remark 3.11.1 is indicative of the differences in the mesh restrictions that are required. We have the following theorem, which limits the rate of convergence.

as

Theorem 3.12.1. Let (A,) be a sequence of meshes with II A N II -+ 0 N -+ co and R = SUP, RAN < CO. Let f ( x ) be in C4[a, b] and p > 0. If

f(4= s ~ N ( f 4; + O(ll A N l14+p)

(3.12.3)

uniformZy for x in [a, b], then

Proof.

Just as in Section 3.1 1, we can show

Thus, for any E > 0 and N sufficiently large, 1 8;[xEl, xiy 1 < E, since 8 $ A N [ x E 1xiN] , = 0 for all N . I n addition, we know that, for some xi, in xiN], 8f4[xLl, xiN] = f ( 4 ) ( x i N ) . Since f(4)(x) is uniformly continuous on [a, 61, it follows that

[%El,

for 11 A , Ij sufficiently small. This proves the theorem.

3.13.

97

HILBERT SPACE INTERPRETATION

3.13. Hilbert Space Interpretation T h e class X 2 ( a ,b ) defined in Section 3.1 is a Hilbert space under the inner product

.r”

(f,g) = f ” ( 4 g W dx

(3.13.1)

if one makes allowance for the pseudo-character of the inner product. For any mesh d: a = xo < x1 < * * . < x N = b, the family, FA , of all cubic splines on d is clearly a linear subspace of Y 2 ( a ,b). Since a nonperiodic cubic spline on d is determined by its values on d and the values of its first derivative at x,, and xN (which can be taken as its 3, or N 1 if we defining values*), this subspace has dimension N allow for the fact that we are actually interested in the equivalence classes modulo the two-dimensional subspace of linear functions on [a, b ] . I n this sense the subspace, P A , of periodic cubic splines has dimension N - 1. Since both FA and PA are finite dimensional, they are closed subspaces of X 2 ( a ,b) with respect to the norm

+

+

(3.13.2)

determined by (3.13.1). If d l C A , , then FAlCFA, and PA,C PA,. T h e Gram-Schmidt orthogonalization procedure allows us to introduce an orthonormal basis into either FA or PA If d, we denote by [FAN+l - F A N ] the subset of FAN+, consisting of splines SdN+,(x)which vanish at the mesh points of d, and whose first derivatives vanish at x = a and x = b. Thus, [FAN,,- FAN]is the subset of FAN+1 consisting of splines whose defining values on d, vanish. Similarly, [PdN+,- PAN]is the subset of PAN+1 consisting of splines whose defining values on A , vanish. We recall that, if (Vi}(i = 1, 2, ...) is a sequence of mutually orthogonal subspaces of X 2 ( a , b), then the infinite direct sum .+

v, = v,@ v2@

..’

is the smallest linear subspace of Y 2 ( a , b) which contains all the finite direct sums v,CB v,@ ... @ v, ( N = 1,2,...)

* We assume, here and in the remainder of Chapter 111, the defining values to be the quantities ay: fly: and yt in the Y vector [(3.7.7.7) or (3.7.7.13)] appropriate to the spline under consideration. + In this section we speak of splines rather than equivalence classes of splines in order to simplify the arguments.

+

98

111.

INTRINSIC PROPERTIES OF CUBIC SPLINES

and is closed with respect to the norm (3.13.2). An element v in V , has a unique representation m

v

=

pi,

i=l

where vi is in Vi(i = 1, 2, ...). Let {A,} ( N = 1, 2, ...) be a sequence of meshes on [a, b] with A , C A,,, . Then, if N # [FA, - FAN-,] and [FAfl - FAN-1 - ] have > N ; then if S(x) is in only the zero spline in common. Assume [FAN- FdN-J,it is a spline on A,; if S(x) is in [FAR- FAN-l],its defining values on B vanish. Consequently, S(x) is a spline on A N whose defining values on A N vanish; therefore, S(x) vanishes identically. We observe that .Xp2(a, b) is not a closed subspace of X 2 ( a ,b ) but is dense in X 2 ( a ,b) (cf. Section 6.14). We now raise the question, “Given a sequence of meshes, when are the associated sequences of linear spaces {FAN}and {PAN]such that

m,

m

m

FA,

E

FA, CB

C @ [FdN -FAN-l] N=2

= y 2 ( a ,b),

(3.13.3)

I n Section 3.14, we obtain an important set of sufficient conditions for this to be the case. We note that, since FA, C Z 2 ( a ,b), it is sufficient to show that, for any f(x) in X 2 ( a ,b) and an orthonormal basis {vi(x)} for FA , W

&j

(if c ( f , 1 N

Vibi

-

i=l

= 0.

(3.13.5)

Similar remarks apply to PAWand Xp2(a, b), since the latter is dense in X 2 ( a ,b). We show in Section 3.14 that the component spaces in (3.13.3) are mutually orthogonal so that the decomposition (3.13.3) is defined; we do the same for the decomposition (3.13.4).

3.14. Convergence in Norm I n Section 3.14, we establish two important theorems, the first of which is a convergence theorem that could have been included in Section 3.8.

3.14.

CONVERGENCE IN NORM

99

Theorem 3.14.1. Let {A,} ( N = I, 2, ...) be a sequence of meshes with A , C A,,, and I/ A , I/ -+ 0 as N -+ CO. Let f ( x ) in &-,(a, b ) be given, and let {SAN( f ; x)} ( N = 1, 2, ...) be a sequence of splines of interpolation to f ( x ) on the meshes A , such that one of the conditions, (a)f(x) - S,,(f; x) is of type I' ( N = 1, 2,...), (b) S,,(f; x) is of type II' ( N = 1, 2 ,...), (c) f ( x ) and S d N ( f x) ; are in &-:(a, b ) ( N = 1, 2, ...), is satisjied. Then lim

N+m

Ilf-

'dN,fIl

z==

(3.14.1)

O.

Proof. If Nl < N , , then the minimum norm property implies that

II SdN1.f /I < ll s,,,,

(3.14.2)

I19

since S, ( f ; x) is in X 2 ( a ,b ) and SAN1( f;x) is the spline of interpolation Nz to S d N Z ( x) f ; on A N l satisfying one of the conditions (a), (b), or (c). T h e 11) therefore is monotone increasing sequence of real numbers (11 and is bounded above by 11 f 11, the latter property again being a consequence of the minimum norm property. It follows that (11 S d N ,I\}, is a Cauchy sequence of real numbers. By the same argument used to establish (3.14.2), we know that the first integral relation applies to S,,(f; x) and S,,(f; x). As a result, we have

I/ S A N + ~, ~SdN,fIl2 = II SoN+,,r I!,

- /I SdN,rl12

(P = 1, Z-..), (3.14.3)

which implies that { S i N ( fx)> ; is a Cauchy sequence in L2(a,b). Since L 2 ( a ,b) is complete, we can find a function g ( x ) in L2(a,b) such that lim

N+m

b a

{g(x) - SJN(f;x)}, dx = 0.

(3.14.4)

Let (3.14.5)

then

by Schwarz’s inequality. Consequently, (3.14.4) implies that for each x in [a, b]

100

111.

INTRINSIC PROPERTIES OF CUBIC SPLINES

since lim I f ’ ( x ) - S i N ( f ;x)

N-tm

1

(3.14.7)

=0

uniformly for x in [a, b] by Theorem 3.8.1. Indeed, since

If’@)

-

G(x)

1

< If’(4- Si,(f; 4 1 + I G ( 4 - Si,(f; 4

I7

f ’(x) must be identical with G(x).This, however, implies that f “(x) = g(x) a.e., which establishes the theorem. Theorem 3.14.1 and Lemma 3.14.1 below provide the major tools needed to demonstrate the validity of the decompositions (3.13.3) and (3.1 3.4).

Lemma 3.14.1. Let A , C A , be two meshes on [a, b]. If SAl(x)and SA2(x)are splines on A , and A , , respectively, such that SA2(x)vanishes on A , , then (SA1, SAz)= 0 if SAl(x)is of type 11’, or SA,(x)is of type 1’, or both SAl(x)and SA2(x)are periodic. Proof. Let A , be defined by a = x,,< x, integrate (SAl, SA2)by parts twice, then

<

1..

< x,

=

b. If we

Theorem 3.14.2. Let {A,} ( N = 1, 2,...) be a sequence of meshes with A , C A,+1 and 11 A , 11 + 0 as N - t co. For each N , let FA, be the linear space of cubic splines on A N , PA, the subspace of periodic splines, and F;, the subspace of type 11’ cubic splines. Then we have x2(a,

b,

FAl

@

m @[FAN

-FAN-)

FAm >

(3.14.8)

N=2

,X2(a, 6)

‘A,

@

W

N=2

@ [‘A,

- ‘AN-1]

‘

Am’

(3.14.10)

3.15.

101

CANONICAL MESH BASES A N D THEIR PROPERTIES

Proof. We prove only (3.14.8), since the proof of (3.14.9) and (3.14.10) are completely analogous. By definition, FA, is closed, and consequently we have the decomposition s 2 ( a ,b) = FA,

0GA, ,

(3.14.1 1)

where GAm denotes the orthogonal complement of FA,. Since FA C Z 2 ( a ,b), we need only prove that f(x) is in FA, if it is in Z 2 ( a ,b). Lg { S A N ( fx)} ; ( N = 1, 2, ...) be the sequence of type I splines of interpolation to f(x) determined by {A,) such that f(x) - SdN(f; x) is of type I’for each N . Then, by Theorem 3.14.1, lim

Nim

1l.f - sA,,f

11

= 0.

(3.14.12)

This proves the theorem, since FA, is closed, and Lemma 3.14.1 establishes the orthogonality of the component spaces.

REMARK3.14.1. I n establishing (3.14.9) and (3.14. lo), we choose the f;x)) to consist of type 11’ splines or periodic splines, sequence {SAN( respectively. I n (3.14.9), for each N , [F& -&’;,-,I is the family of type 11’ splines on A , whose defining values on ANpl vanish. In the periodic case we also use the fact that Z j ( a , b) is dense in Z 2 ( a ,6). We consider this in greater detail in Section 6.14.

3.15. Canonical Mesh Bases and Their Properties Let {A,} ( N = 1, 2, ...) be a sequence of meshes on [a, b] with A N C A,,, . We assume that we are given an orthonormal basis for FA,, and we extend this to an orthonormal basis for FA, by constructing an orthonormal basis for the orthogonal complement, [FA,-F,J, of FA,with respect to FA, . T h e construction yields for every N an orthonormal basis for [FA, - F A J .T h e same method yields with slight modifications orthonormal bases for [Fi, - FiJ and [PA, - PA,]. Theorem 3.14.2 then shows that this construction provides explicit orthonormal bases for X 2 [ a ,61 if I/ A , /I -+ 0 as N A 00. We call mesh bases these orthonormal bases for [FA, - FAJ, [F;, - F i J , and [PA,

- PAl].

Consider the set 42 of all distinct mesh points determined by the sequence of meshes { A N } excluding those that comprise A , . Since M is denumerable, let it have a specific enumeration M

= {PI , P ,

,...}.

(3.15.1)

102

111.

INTRINSIC PROPERTIES OF CUBIC SPLINES

I n the case where only a mesh basis for [FA, - FAJ is desired, M has only a finite number of elements and is denoted by M,. Let A,, be the mesh obtained by inserting the point P, into the mesh A , , and let p,(x) be the type I’ spline on A , , such that pl(P1) = 1 and pl(x) vanishes on A , . If Ali is the mesh obtained by inserting Pi into the mesh = A , ) , then pi(x) is the type I’ spline on A I i such that pi(Pi) = 1 and pi( x) vanishes on A l , i - l . Lemma 3.14.1 assures us that the sequence {pi(x)) ( i = 1, 2, ...) consists of mutually orthogonal type I’ splines and by the manner of its construction is a basis for [FA,,, - F A J , where Z S FAl

@ LFA1l

-FAl]

@ LFAlz

-FA1ll

@ *”

*

Suppose (3.15.2)

the resulting sequence { d i ( x ) )( i = I , 2, ...) of orthonormal type I’ splines is a basis for [FAl,m- FAJ. By proceeding in a similar manner, but requiring that p i ( x ) be of type 11’ or periodic, we are led to mesh bases - FiJ and [PA,,,- P A J , respectively, where the additional for [FL1,, definitions needed are obvious. These bases are not unique, since the construction depends on how M is enumerated; moreover, they are not the desired bases for [FA, - FAA, [ F l , - FiJ, and [ P A - PA,]. If, however, the process exhausts the points of A , for each N before any points not in A , are enumerated, then we obtain the desired mesh bases. We single out one natural way of making this type of enumeration and refer to the mesh bases generated by it as canonical mesh bases. T h e enumeration employed in their construction consists of ordering from left to right for each N the mesh points that are in A , but not in A , - , . Thus, when a mesh basis is canonical, its construction is completely defined and depends on a given sequence of meshes. Any mesh basis is canonical, however, with respect to the auxiliary sequence of meshes {A,,i} ( i = 0, 1,...) used in its construction.

Lemma 3,15,1, Let { d i ( x ) ) ( i = 1, 2,...) be a canonical mesh basisfor [FA, - FAJ, [F;, - F i J , or [ P A , - PAJ determined by a sequence of meshes (A,) ( N = 1, 2,...) with A , C A,,, . Let {Ali) (i = 1, 2,...) be the related sequence of meshes used in the construction of {di(x)). Then

I ,pyx)

j

<

21‘2

j j A,$ jj(3-2a)/2

( a = 0, 1 ;

i

=

1, 2 ,...).

(3.15.3)

3.16.

103

REMAINDER FORMULAS

Proof. Let Ali be determined by a = xoi < xli < < x;, = b. For any x in [a, b], we can find an interval xjP2 x xii such that d i ( x j - 2 ) = di(xii) = 0, since di(x) vanishes on Al,i-l. Therefore, we can ~?], such that a;(xij) = 0. I t follows with the help find an xij in [ ~ j - x of Schwarz’s inequality that

< <

1 d:(x) 1 ,<

Ix 1 2

< / / di(x) / / - 1 x - x i j

1 1 dx 1

.;(x)

)1/2

,< 2 l I 2 / / dli

J

If x is in xii], then either di(xi-l) = 0 or di(xji) = 0. Consequently, we can assume with no loss of generality that di(xi-l) = 0. Thus,

I +)

1

<

Ti I

b i w

I dx G 2112 11 dli 1 1 3 / 2 ,

Xi-1

which concludes the proof. Observe that for each N there is an i, such that A , = Al,{,; consequently, if I/ A , /I -+0 as N - t 00, then 11 dli I/ -+ 0 as i-t CQ. If we assume that Ij Ali 11 -+ 0 as i -t CQ at a reasonable rate, we obtain the following theorem.

Theorem 3.15.1. Let { d i ( x ) ) (i = 1, 2,...) be a canonical mesh basis such that 11 A I i 11 K / if o r some K > 0 that is independent of i. Then there exists a real number 3 > 0 such that

<

‘f (d8(x)>2< B < co.

i=l

Proof. Lemma 3.15.1 justifies the following calculation, which proves the theorem: i=l

i=l

i=l

REMARK3.15.1. T h e conditions imposed on { A l i } in Theorem 3.15.1 can normally be verified from the properties of {A,} due to the close relationship existing between the two sequences of meshes. 3.16, Remainder Formulas

< x, = b be given. Let f ( x ) in ,X2(a, b ) and A , : a = xo < x1 < In addition, let SAl( f;x) be the type I spline of interpolation to f ( x ) on

104

111.

INTRINSIC PROPERTIES OF CUBIC SPLINES

A , such that f ( x ) - S A l ( f x) ; is of type 1'. We now investigate the remainder R ( x ) f(%) - sAdh (3.16.1) By adding new mesh points at the midpoints of old mesh intervals, we can define a sequence of meshes {A,} ( N = 1, 2, ...), where d, C A,,, and 11 A , 11 --+0 as N - + 00. I n each A , there will be (2N-1* K ) 1 mesh points. If we form the canonical mesh basis {oi(x)} (i = 1, 2, ...) for [FA,-FAA and its associated sequence of meshes {Ali} (i = 1,2, ...), then d1,2N-zK = ( N = 2, 3,...). (3.16.2)

+

4,

Since d,

(i = 2N-2K,2N-2K

C dli L A N + ,

+ 1,..., 2N-'K),

it follows that

<

<

in view of the fact that i 2N-1K implies that ( + ) N - l Kli. Consequently, the conditions of Theorem 3.15.1 are met so that

c m

S(x) =

(.i(.)}2

i=l

< 3 < co

(3.16.4)

for some positive real number B. Let n

Kn(x, t )

=

1

cii(X)cii(t)

(n = 1, 2,...) ;

(3.16.5)

i=l

then

by the orthonormality of the functions oi(t). I n view of (3.16.6) and (3.16.4), {Kn(x,t ) } (n = 1, 2, ...) is a Cauchy sequence in L2(a,b ) for each x in [a, b ] . T h e Schwarz inequality shows that it is also a Cauchy sequence in L ( a , b) for each x in [a, b]. Let K(x, t ) denote the common limit; then, by (3.16.4) and (3.16.6), the integral J b K ( x , t ) 2 dt is uniformly bounded with respect to x. Thus, K(x, t ) is inLZ(a[a, b] x [a, b ] ) .

3.17.

TRANSFORMATIONS DEFINED BY A MESH

105

It also follows that

= lim n+m

= lim n-m

1

b

bi(x)

.

2=1

a

b:(t)f”(t)

dt

n

C .

,f)di(x)

(di

2=l

We summarize these results in Theorem 3.16.1.

Theorem 3.16.1. Let f(x) in X 2 ( a ,b) and A , : a = xO < x, < -.. SA1( f;x) be the type I spline of interpolation to f(x) on A , such that f ( x ) - SAl(f;x) is of type I’. Then there exists a kernel K(x, t ) in L2([a,b] x [a, 61) which is in L ( a , b ) and L2(a,b) for every x in [a, b] and

< xk = b be given. I n addition, let

R(x) = f ( x ) - SAl(f;x)

J a K(x, t)f”(t) b

=

dt.

(3.16.7)

REMARK3.16.1. Similar kernels can be obtained for type 11’ and periodic spline approximations. Observe that the kernels are independent O f f(4.

3.17. Transformations Defined by a Mesh Sard [1963] has used remainder formulas of the form (3.16.7) extensively in the analysis of a wide variety of approximations. T h e existence of kernels K(x, t ) for the case wheref”(x) in (3.16.7) is replaced by an arbitrary derivative has been established under very general conditions. T h e initial results date back to Peano [1913]. I n Chapter V, we obtain kernels for spline approximations where derivatives other than the second derivative play the principal role. A more general point of view has been taken by Greville [1964], who has investigated the existence of such kernels whenf”(x) is replaced in (3.16.7) by a linear differential operator L of order n. Greville [1964] also established, for a linear differential operator L and a family of transformations of the form (3.17.1)

106

111.

defined by K a = xo < x1

<

INTRINSIC PROPERTIES OF CUBIC SPLINES

+I

functions ui(x) in X n ( a , b) and a mesh A :

< x k = b, the result that if

then (3.17.3)

is minimized for each x in [a, b] when the functions ui(x) in (3.17.1) are selected such that Tf(x) is a generalized type 11’ spline of interpolation tof(x) on A . This is a partial extension of Schoenberg’s [1964b] results on the approximation of linear functionals to the generalized setting. I n Chapter VI, we give a more complete extension of Schoenberg’s results to generalized splines. A slightly more general form for the transformation T defined by (3.17.1) is given by

where the additional functions bo(x) and bl(x) are also in X n ( a , b). We restrict ourselves to the case n 2, regard X n ( a , b) as a Hilbert space with functions differing by a constant identified, and inner product =3

interpret T and R , as linear mappings of Y 2 ( a ,b ) into itself, and ask, “Under what conditions on the functions ui(x) (i = 0, 1, ..., K ) , bo(x), and bl(x) will 11 R , /I be a minimum ?” We can represent X 2 ( a ,b) as X 2 ( a ,b) = FA @ Gd

,

(3.17.6.1)

where GA = [%‘(a, b) -3’41

(3.17.6.2)

is the orthogonal complement of FA and is identical with the set of all functions in X 2 ( a ,b) which vanish on A and have first derivatives that vanish at x = a and x = b. Thus, forf(x) in G, , RTW

=

f(4,

(3.17.7)

3.18.

A CONNECTION WITH SPACE TECHNOLOGY

107

which implies 11 R,jl 3 1 for all allowable T . A constant function f ( x ) is equivalent to the zero function, and so we must have Tf equivalent to 0 in this case. Consequently, if we want T to be defined on ,X2(u, b) with the indicated identifications, we must require that

c4 4 k

i=O

be a constant function of x. Let the functions u i (x ) be chosen as the cubic type I’ splines on d such that u i ( x j ) = aii, bo(x) as the cubic spline on d which vanishes at every mesh point of d but has a unit first derivative at xo and a zero first derivative at xk , and b,(x) as the analogous spline to bo(x) but with prescribed values of the derivatives interchanged. Then T f ( x ) is the projection of f(x) on FA . Moreover,

is a constant function of x, in fact identically 1. I n this case, the linear mappings T and R, are projections whose associated linear subspaces FA and G, are orthogonal complements. As a consequence,

II TI1 = II RTII = 1.

(3.17.8)

For this choice of T , not only is T a projection, but, as a projection, it is larger than any other projection in the class of transformations under consideration, since the null space of any other projection of the form (3.17.4) must contain G, . If T is not a projection, then there is a better approximation in T [ X 2 ( a b)] , tof(x) than Tf(x). If T is a projection and T I > T is also a projection, then T,f(x) is at least as good an approximation tof(x) as TF(x). Here the measure of goodness of an approximation Tf(x)tof(x) is in terms of the smallness of 11 RTf 11. With slight modifications of (3.17.4), we can define two related classes of transformations where type 11’ splines and periodic splines, respectively, play the same role that type I splines play for this class of transformations.

3.18. A Connection with Space Technology I n order to maximize the payload delivered by a rocket, we must minimize the integral of the square of the applied acceleration (Seifert

108

111.

INTRINSIC PROPERTIES OF CUBIC SPLINES

[1959, 10-2-21). I n a gravitation-free field, this is equivalent to minimizing the integral of the square of the total acceleration. If both initial and terminal position and velocity are prescribed, as well as the time of flight, Theorem 3.4.3 tells us that the resultant trajectory has coordinates, each of which is a cubic with respect to time. This is in agreement with standard engineering analysis (Seifert [ 1959, 10-3-11). If intermediate positions are prescribed for times other than the initial and terminal time, the solution is a trajectory whose coordinates are splines with respect to time. Again this follows from Theorem 3.4.3. We observe that the applied acceleration need not be assumed continuous, only square integrable. I n standard analysis, the optimization is restricted to acceleration profiles that have second derivatives with respect to time in order to accommodate a double integration by parts. I n the proof of Theorem 3.4.3, only the spline is differentiated, not the representative function from the class X 2 ( a , b) over which the optimization takes place.

CHAPTER IV

The Polynomial Spline

4.1. Definition and Working Equations

It is natural to attempt to extend the concept of the cubic spline to curves that are composed of segments of polynomial curves of an arbitrary given degree and to investigate extensions of the properties ascribed to the cubic spline in the previous two chapters. T h e purpose of this chapter is to introduce polynomial splines and to consider their algebraic properties. I n the following chapter, a detailed investigation of intrinsic properties of polynomial splines of odd degree is presented. T h e first significant item that one encounters in the extension to polynomial splines is that there is an essential difference between splines of even and odd degree. One finds, for example, that polynomial splines of even degree interpolating to a prescribed function at mesh points need not exist. For this reason, the definition of an odd-degree spline of interpolation which does, in fact, yield the expected extensions of cubic spline properties must be modified for splines of even degree. We consider first the number of degrees of freedom involved. For polynomial splines of degree 2n - 1 on a mesh A : u = x0

< x1 <

***

< x N = b,

there are 2nN constants to be determined. Requiring derivatives of orders 0, 1,..., 2n - 2 to be continuous at each interior mesh point 2n - 1. accounts for (2n - 1)(N- 1) degrees of freedom, leaving N We require interpolation at the N + 1 mesh points and impose n - 1 end conditions at x = x,, and at x = x N . For splines of degree 2n, we have (2n l)N degrees of freedom. Requiring continuity for derivatives of orders 0, 1, ..., 2% - 1 at interior N are n mesh points yields 2n(N - I ) conditions. T h e remaining 2n end conditions at x = xo and at x = x N with one condition for interpolation in each interval. A natural procedure to be used for N data points is to take mesh points midway between data points (points of interpolation) with end intervals bisected by the first and last data points.

+

+

+

109

110

Iv.

THE POLYNOMIAL SPLINE

I n this chapter, no attempt is made at completeness in the discussion

of even-degree splines. They are introduced principally in situations

in which they combine with odd-degree splines to help clarify the total picture. For a polynomial spline S,(x) of degree k on a mesh A , then, we require interpolation to a prescribed functionf(x) at the points of the mesh if k is odd, and at interval mid-points

(i = 1,...,N ;

< ti < xi)

xi-1

when k is even. Periodic splines of degree 2n - 1 or 2n on A satisfy, in addition, the requirements SLq'(x0+) = SLq'(xN-) for q = 0, 1,..., 2n - 2 and q = 0, 1,..., 2n - 1, respectively. From the Taylor theorem with integral remainder, we have

For the spline of degree 2n - 1, we employ this expansion in the form

0,

where, for m

xT=O, = xm,

x GO, x

> 0.

I n this notation, the unit step function with step at x = 0 is x.! When N >, 2n - 2 and n - 1 \< i N -n 1, taking centered divided differences,

<

x

(Xi

- t ) -+3

dt,

+

(4.1.2)

inasmuch as SJz"-2)(x) is linear on each interval (xi-1,xj), and the first term of (4.1.1) is a polynomial of degree 2n - 3 in x i . We have

4.1.

111

DEFINITION AND WORKING EQUATIONS

d(xi - t),"/dt = -.(xi - t):-l for n 3 1. Thus integration by parts gives, for the right-hand member of (4.1.2),

-

-

Mi - Mj-, ( x i - t)";"-' hi (2n - 2)(2n - 1)

[

1 (2n

-

2)!

sy-zn/r,

I ( Xi

- X,)y-2 + (Xi -

.,,y--'

- (Xi

(2n - l)hl

-X

y-1 O +

1

where Mi= S~zn-2)(xi). T h e function S;(x - xj)Tp1 is identically zero for x xjP1 and on x 3 xj+, coincides with a polynomial of degree 2n - 3. Thus, 6 F - 2 S 5 ( ~ i - x 3. )f ~ ~ -= ' 0 for i < j - n and for i > j n. If i 3 1, + - (xi - ~ , ) ~ - ~ ] / h , (2 n1) is a then ( x i - X,)Y-~ [(xi- x 1)2n-1 polynomial of degree 2n - 3 in x i . Thus,

<

+

+

8?-'{(xi

-

~ ~ ) 2 ++- [(.xi ~

-

xl)Y-' - (xi

-

~ , , ) y - ~ ] / h ~( 21)) n =0

if i

n.

For i = n - 1, we may rewrite the coefficient of M, as 1

(2n

-

8;n-2

2)!

-

[

( X i - xl)Y-'

(2n

e2+

(2n - l)! n;fl;2(h,

- ( X i - X1)Zn--1 -

*.*

l)h,

+ hj)

Ii+, (4.1.4.1)

(4.1.4.2)

112

Iv.

For n - 1

THE POLYNOMIAL SPLINE

< i < N - n + I , we may write (4.1.2) as i=i+n-1

C

AijMj

= SyP2S~(xi),

(4.1.5)

i=i-n+l

where A,.-

23

1 (2n - l ) !

(hj

+ hj+l) 87-2 q x i - x j ) y l ,

(4.1.6)

provided we define, in accordance with (4.1.4),

I n the periodic case, where we designate xiPN = xj - ( b - a ) and xjtN = xi ( b - u) and employ the periodic character of SA(x), the term in M,, in (4.1.3) drops out, and (4.1.5) is valid for each point of the mesh. We note that, since the quantities Aij are independent of S,(x), we may replace SA(x)in (4.1.5) by the function x ~ which ~ - is ~ itself a spline of degree 2n - 1 on the mesh. Thus, (4.1.5) yields

+

(2n - 2)!

i+n-1

C

(n - 1

Aij = 1

i=i-n+l


-

n

+ 1).

(4.1.7.1)

t 1).

(4.1.7.2)

Similarly, if we replace Sd(x) by x ~ ~ we - ~ obtain , (272 - l)!

i+n-1

1

xjAij

j=i-n+l

itn-1

=

1

(n - 1

xi

j=i-n+l


-

n

I n the periodic case, Eqs. (4.1.7) are true for all i. An additional important property is that each A,,j 3 0. I n fact, if we consider the function +(x)

=

(x - x.3+1 +

-

(x - x.)2n--1

-

hj+l

we find that d(”)(x) = 0 ( p +(2n-2)

3 +

=

(x - x.)2n-1 - (x - x. y 1 3 + 3-1 + , hi

0,1,..., 2n

(x) = (2n - 1)!(x - xj-l)/hj

-

in

= (2n - l ) ! ( ~ ~ +~ ~) / h ~ +in ~ =0 for x

2) for x <

that

< x < xj , xi < x < xj+l,

Thus, the first 2n - 2 derivatives are nowhere negative, and all the divided differences of order 2n - 2 in (4.1.6) are consequently nonnegative.

4.1.

DEFINITION A N D WORKING EQUATIONS

113

For periodic splines and for nonperiodic splines when n-1


the spline equations are given by (4.1.5). For nonperiodic splines, there are, in addition, the 2n - 2 relations resulting from spline end conditions. I n determining these, we restrict our attention to splines of certain types. By a simple spline of degree 2n - 1, we mean a polynomial spline of degree 2n - 1 which is in C2n-2[a,b]. Furthermore, we say a polynomial spline of degree 2n - 1 and in general a function (when the value of n is clear) is of type I’ if its first n - 1 derivatives vanish at x = a and x = b. Two functions are in the same type I equivalence class if their difference. is of type 1’. I n this same context (again when the value of n is clear), we say that f ( x ) is of type 11’ if f ( ” ) ( a )= f ( p ) ( b )= 0 for p = n, n 1,..., 2n - 2. Two functions are in the same type I1 equivalence class if their difference is of type 11’. Also, a spline is of type I if it is represented in a manner in whichf(p)(a) and f ( p ) ( b ) ( p= 1, 2 , ..., n - 1) enter explicitly into the representation as independent parameters, and a spline is of type I1 if it is represented in a manner in which f ( p ) ( a )and f ( p ) ( b ) ( p= n, n 1,..., 2n - 2 ) enter explicitly into the representation. I n these situations, the dependence upon the quantities f ( p ) ( a ) and f @ ) ( b ) is to be linear. I n order to impose the end conditions for splines of types I and I1 at x = a, we form divided differences from (4.1.1). Let [xo,x1 ,..., xiIk and f x o , x1 ,..., xj]: denote the divided differences of xk and x + ~ ,respectively, over xo,x1 ,..., xi. Then from (4.1.1) we obtain, employing the method used in obtaining (4.1.3), the equations

+

+

+ [xo -

x1

, 0, x2 - x1 )...,xq - x 1 1 y

-

(2n - l ) h ,

[O,

XI -

xo

,..., x p - x012+n-1\ ! (4.1 .S)

q = 1, 2 ,..., 2n

-

3 , where d,(X)

= S?(X - Xj)”;”-’.

114

Iv.

THE POLYNOMIAL SPLINE

T h e last sum in (4.1.8) is taken over the index set I, ..., q inasmuch as - 1. For type I splines, we eliminate the unknown quantities S g ) ( a ) (4 = n, n I , ..., 2n - 3 ) and obtain n - 1 equations in M , , M , ,..., M 2 n - 3 . For type I1 splines, we take Eqs. (4.1.8) for q = n ,..., 2n - 3 and obtain n - 2 relations in M,, ,..., M2n-3which, together with the specification of M,, under the type I1 condition, again constitute n - 1 equations. A similar set of n - 1 end conditions is obtained at x = b using in place of (4.1.1) the relation

dj(xk)= 0 when k < j

+

+

I - 2(n - 1) standard spline equations of the We have, then, N 1 equations form (4.1.5) together with 2n - 2 end conditions, in all N in the quantities M,, , M , ,..., M N . When N < 2n, then the standard spline equation (4.1.5) does not apply, and the determining equations are obtained solely from (4.1.8) and its counterpart associated with (4.1.9). I n this connection, we note that a necessary condition for the existence of a type I1 spline is N 3 n - 1. If we were to integrate

+

< <

on x, x x1 , n - 1 times using S;*'(a) =f'*)(a) (q = n,..., 2n - 2), we could determine Si*)(x,)(q = n,..., 2n - 2) except for the single x x2 would give parameter M , . Integration of Sizn-2)(x)over x1 S$'(x,)(q= a ,..., 2n - 2) in terms of the parameters M I , M , . Finally, xN would yield Siq)(b)(q = n,..., 2n - 2) integration over xNp1 x in terms of M , ,..., M N . Since the quantities Si*)(b)(q = n ,..., 2n - 2) are prescribed, there are N - n I degrees of freedom remaining at this point, and so N n - 1. Carrying out n further integrations 1 n =N 1 degrees of freedom and hence, now gives N - n conceivably, the possibility of having So(xj) = fj (prescribed),

< <

< <

> + +

+ +

j = o , l , ..., N.

T h e complete determination of the spline next requires the values of the derivatives SL*)(xj)(q = 1,..., 2n - 3 ; j ' = 0, 1,..., N ) from the values of the quantities M i and the values SA(xj) to which the spline interpolates at the points of d. For xi-, x < xi, direct integration of

<

(4.1.10)

4.1.DEFINITION

AND WORKING EQUATIONS

I15

gives

+i(p k=l

Bkj

(p

.( - xj)”-k

=

1, 2 ,..., 2n - 2 ) ,

(4.1.11)

+

(4.1.12)

-

where

B k j = S Y - 2 - k ) (xj) - Mj[hjk/(k l)!].

+

< <

When n - 1 j N -n 1, taking divided differences in (4.1.l), with a replaced by x i , gives a determining set of equations for the Bki . For p even, we have

Fj(Xi) =

1 (2n - 3)!

j;;sp-z)

hy-2 + M i(2n - l ) !

(t)(Xi

- q2n-3 dt

+ (2n -1 l)! 1 Mm(h, + L,,) 6 3 % - %P1 i-1

m=j+l

( - h .Z + l )2n--2 1 + M i (2n - I ) ! (2% - I ) !

For

(i > j ) ,

c ~ m ( h m+ h,+l)

j-1

m=i+l

,-..*Xj-1

9

*i+1

9..*!

G ( X , - Xm)2n-1

(i < j ) .

p odd, we employ the divided difference SA[Xj-(Zl+1)/2

(4.1.14)

(4.1.15)

Xi+(P+l)/Zl.

T h e resulting equations in the quantities Bk,i(K = 1, 2 , ..., 2n - 3 ) have a triangular coefficient matrix, and the method of solution is evident. We turn now to an alternative procedure for constructing the standard Such a procedure is desirable because the algebraic spline relation (4.1.5). is quite evaluation of the divided differences Sp-’8;(xi tedious. We require first the polynomial p(.) of degree 2n 1 in u such that

+

Iv.

116

THE POLYNOMIAL SPLINE

p(u) and its first n derivatives take on prescribed values at cr = u =

1:

p'"(0)

=p

y,

p'k'(1)

Write p(u) in the form

p(.)

=p ,

+ p ; + (1/2!)p32 + 0

***

=p

y

(K

0 and

,...,n).

= 0, 1

+ (1/n!)pk)an+

Then, imposing the prescribed conditions at u of equations

&"+l

=

+ + ***

1 gives the system

k-0

c n

p; -

pF'/(k - l)!

k=l

pi -

pF'/(k - 2)! k=2

(4.1.16)

where k ~ i )is the factorial function k(k Multiply on the left by the matrix

T h e resulting matrix on the left is

-

1)(K

-

2)

... (k - j

+ 1).

4.1.

117

DEFINITION AND WORKING EQUATIONS

The inverse of this matrix is -1 1

1 2!

_ _1

...

1 2!

...

3!

-1

0 _l . 2!

1

0

0

1 -( - 1 )

0

0

0

0

0

0

(-

1)s-1

(n - I)!

2 ! (n - 2)!

1 _ (-1)"-3 . ... _ ___

1

3!

-

n!

1 (-1)n-Z ... -~

2!

_1 .

(__ elln

3! (n - 3)!

1 _ (-1)-4 ... _ __

4! (n - 4)!

0

...

1 n!

-

Furthermore, we have

-

1

0

v=

0 0 0 -0 0

1 1 -

2! 1

1 3! 4! 1 1 2! 3!

1

1

0 0

0 0

1

... ...

(n - l)!

1

- ... 2!

0 0

... 1 ... 0

(n

-

2)!

1 1

Thus, (4.1.17)

118

Iv.

We find that

THE POLYNOMIAL SPLINE

is equal to (yij),

where (k) = 0 if m < 0 or m > n, (g) being the binomial coefficient "C,, (:) = 1. T h e product B d Y , however, is much simpler in form and suffices for the determination of the polynomial:

...

)i 1

-i

1

1 n+2 n -1 2 4 + 2 m i L 1 ) 3 n + 2 1 (n - l)! 2 n - 2

1

0 n + l

11

(n - l)! 0

-do)(

-4

-1 1 n + l T i l L l ) 1 2 n + l - 2)

n

TU

-4 li 1 Y!dX (-1)m

(n - l)!

n + l 4

n + 2 0

n + l 0

(-1)W

. (4*1J8)

11

We may write

where 1

T%,&(IJ) =--k! k ! j=o

.

)

un+j+l.

(4.1.20)

4.1.

DEFINITION AND WORKING EQUATIONS

119

Consider now the component polynomial of S d ( x ) on the interval xi-1 x xi. Designate this by Sd,j(x):

< <

Then, for q

=

(4.1.21)

n,..., 2n - 1, we have

where

(4.1.23)

I n particular,

(4.1.24)

_ _(-l)n-l _ _ _ (2" k!

2)

(2n - I ) ! .

If we now impose the condition that SLq'(x) be continuous at x i , for q = n, n 1,..., 2n - 2, we have for these values of q the relation Si:i(xi) = SLq,\+l(xj), that is,

+

Iv.

120

THE POLYNOMIAL SPLINE

Denote SiQ’(xj) by SjQ). Then there are n - 1 equations (p

sjy + Tzl*n-l(o)[(-l)n+a-lh;

h;TEl,n-l( 1) -

+ h,Q,l]

= a,...,

2n - 2)

sy-1)

(- l)”+“-lh;+, T$)l*n-l(1) sjy n--2

=-

k=O -

+

{hyT21,k(I)s,’cri Tzl,,(o)[(-l)k+ah;

(-l)k+ah;+l Tzl,k(l)

sjti},

+ h;+1] Sj(k) (4.1.26)

+

< <

at each point xi . When n - 1 i N -n 1, Eqs. (4.1.26) taken at the points , ,..., F ~ + ~give - ~ (2n - 3)(n - 1) relations in the (2n - l)(n - 1) variables Sj’”)(j= i - n

+ 1,...)i +

+

11

- 1;

+

q

1, 2)...,n - 1).

=

Elimination of SjQ)(j =i-n 1,..., i n - 1, p from these gives the spline equation in terms of Sl(n-’)(j = i - n

=

1, 2 ,..., n

-

2)

+ I,..., i + n - I).

For the spline equation in terms of the quantities M i ,we form by means of (4.1.16) the expressions for Sj:i1) and Sjn--l)in terms of Mi-1, M i ,and Sjy1 and SiQ)(q= 0, 1,..., n - 2). We substitute these into (4.1.26) and then carry out the elimination process as before. A relation equivalent to (4.1.5) results. We carry through this construction for the quintic spline as an illustration. T h e functions T2,0(a),T2,1(o),T2,2(a)are found here to be

U-2

T2,2(CJ) =---

2!

2 (303 - 3a4

so that TgO(0) = -60,

Tgl(0)

-36,

Tgz(0) = -9,

T&(l) = -60,

Tgl(1) = -24,

TgZ(1) == -3,

Tifi(0) =

Tifi(0) =

Ti:i(O) =

Ti$(l)

=

360, -360,

=

192,

Tifi(1) = -168,

36,

T:fi(l) = -24.

+ 4,

4.1.

DEFINITION AND WORKING EQUATIONS

121

We have from (4.1.22)

so that

Thus,

Continuity of Sz(x) at xj gives

(4.1.30)

Continuity of Si(x) at xj by (4.1.28) gives h? M . 30

1 +20 (h,”

-

h;+l h;+& Mj - __ 30 Mj+1

(4.1.3 1)

Iv.

122

THE POLYNOMIAL SPLINE

We write Eqs. (4.1.25) and (4.1.26) at xjA1, xi , xi+l and eliminate Si.-2,SjPl, S ; , Si+l, Si+2from these six equations. There results

We have next to exhibit the end condition for the quintic spline. We write these for type I splines, although the methods employed apply equally well to type I1 splines. For x = a, Eqs. (4.1.25) and (4.1.26) are employed to impose continuity on Sl;(x) and S:(x) at x 1 and x 2 . A fifth equation is obtained from the first of Eqs. (4.1.23) (5': is specified). Elimination of S; , Sk, and S; now gives two conditions in M,, , M I , Mz , M3 : h,

Mo

h2 +h3

+ h, I20 [ h , + h, + h3

+

3:;

4.1.

-[$-

h,

DEFINITION A N D WORKING EQUATIONS

123

+1 h, ( 7 - -)I h,

s,- s, s, - so

T h e corresponding end conditions at x = b may be obtained directly from these by symmetry. I n order to represent the spline over the interval (xj-,, xi) once the quantities M i have been determined, it is necessary to determine Si-l , Si , SJ-.l, S;. Then (4.1.21) gives S,(x). If we eliminate SiP2from the two continuity relations at xi-, and then S;,, from those at x i , we obtain two equations in SiPl and Si which yield

Finally, Sip, and 5';:are obtained from (4.1.28). It is apparent that polynomial splines for unequal intervals are cumbersome to work with. Most of the complexity is attributable to the fact that the quantities M iare implicitly determined by the spline equations. We introduce in Section 4.5 quintic splines of maximum deficiency, which on the other hand exhibit little of this complexity and are quite convenient to work with.

Iv.

124

THE POLYNOMIAL SPLINE

4.2, Equal Intervals

Polynomial splines on meshes whose intervals are of equal length are of central importance in the theory of polynomial splines. I n this case, let h = ( b - u ) / N be the interval length, and let d = E - 1 be the difference operator associated with this displacement, Ef(x) = f(x h). We write Eqs. (4.1.5) and (4.1.6) as

+

AP-1)

=

2h SP-' Sf[(; (2n - l)!

-

j)h]";"-'(2n- 2)!hZn-'.

(Note that aj indicates divided difference as heretofore.) Since Sg[(i - j)h]?n-l = 6:[(i - j)h]?-l, we can write this equation as (4.2.2)

If we use now unit intervals for differencing and set

#p)= A"+lE-"(K)n+

(K

= 0,

1 , 2,...,7~

-

l),

(4.2.3)

we have (4.2.4)

Thus,

where Sz signifies the central difference operator E-lA2. for n even are associated with even-degree splines T h e quantities interpolating at mesh points. It is convenient, however, to consider the entire set of these quantities. We note the relationship between advancing differences of xn at x = 0 ("differences of zero"; cf. Freeman [1949, p. 1231) and the (n - 1)th differences of (x):. If we denote dkxm Iz=o by dkOm, then with unit differencing interval we have

4.2.

This relationship is evident in the case m illustration: x

-3

.2

x+3

0

0

Ax+3

A' x+

125

EQUAL INTERVALS

0

-I

=

I

0

3, which serves as an 2

3

4

27

64

37 0

A3 x: A4x:

I n the horizontal box are are 03, do3, A203, d303. Evidently we have

... d"0"

43),

4i3),+A3),

whereas in the diagonal box

... =

+ l$i?2 + + 4:"). **.

Solving these equations for the quantities +p) gives directly (4.2.6). The differences of zero satisfy the recurrence relationship dRO"+l

=

k(d'O"

+ d'c-10").

+tm)satisfy the recurrence relation +im) = (K + 1) +irn-l) + ( m - K) q&l).

(4.2.7)

We show that the quantities

(4.2.8)

Iv.

126

THE POLYNOMIAL SPLINE

(4.2.9)

Now

--(k + 1) ( m - k

-1

j + l

=

+j

(

-(k - j ) m - k - 1 j + l

I[+

1 + j 1

m - k - l + j +j)+(m-kjl

+ 1) (m - kj -+- l 1+ j

)+(m-k-l)(

m - k - l + j j

11,

(4.2.10)

+

and the term in brackets vanishes, since k (”fj) = ( j l)(jk+”).Also, = (7:;). Hence, the left-hand member of (4.2.10) is equaI to m-k-1 + j -(k -i) m - k + -l+j)+(rn-k;l+j (k 1 - i )

(7)+ (&)

[(

)] + +

=-(k-j)(m-~+j)+(k+l-j)(

m - k - l + j

(

).

1

(4.2.11)

Thus, the right-hand member of (4.2.8) is equal to

(4.2.12)

On the other hand, by (4.2.7), we have

(-l)j( m - k - l + j

= i=O

i

) (k + 1

-

j)[Ak+l-3Om-l + A k - j o m ]

4.2. =

(k + 1) Ak+lOm--l

127

EQUAL INTERVALS

+ 2(-1)j k-1

[-(k - j )

c

m-k+i j+l

j=O

Here the last term is equal to zero, and (4.2.8)follows. is given in the following T h e initial part of the array of coefficients table (m refers to row, k to diagonal): 1

m = l

m=2

1 1

k = l

1 4 1

m=3

m=4

k=2

1 11 11 1

k = 3

1 26 66 26 1

m=5

k=4

1 57 302 302 57 1

m=6

m=7

k=O

k=5

1 120 1191 2416 1191 120 1

k=6

For equal intervals, the spline relation for quintic splines, (4.1.27), is seen to collapse to

(4.2.13)

We illustrate also the application to quartic splines on equal intervals. By returning to (4.1.1),we obtain S,(xj) =

I b +I ST(t)(xj 2! a

c2 k1!

- (xj - u ) ~

k=O

-

t)+2 dt

and by a derivation similar to that used for (4.1.5)derive

& (Mj-2

+ 1IMj-1 + 11Mj + Mj+&

=

-

3sj

+ 3s+,h3

'j-,

. (4.2.14)

I n order to formulate the complete equation system for type I and type I1 splines, it is desirable to express (4.1.8) in simplified form.

128

Iv.

THE POLYNOMIAL SPLINE

For uniform meshes, Eqs. (4.1.8) (4

=

1, 2,..., 2n - 3) become

(In the second and third terms on the right, d operates on the index i. Of course, Aq(i)?-2 = dq02n-2.) For the representation of S,(x) and its derivatives on 4, it is much more convenient in the present situation to replace (4.1.11) by the following:

> f o r p = 0, 1, 2,..., n - 1. By the continuity of Si2"-3)(x) and Al,j

= Bl,j

,

Si2n-4)(~) at x i , it follows that

h 2!

-M . j

-

82

- Al,j

2

It is immediately seen by induction that, for p

For p

=

(4.2.16)

=

.

1, 2, ... n

(4.2.17) -

1,

a, we have also h2n-1 M i h2n-3 Al,i (2n - I)! h + (2n - 3 ) !

+ ..* + hAn-,,j = Sj .

(4.2.19)

4.2.

129

EQUAL INTERVALS

Write this system of difference equations as

(4.2.20)

-

h2 2!

82 -_

0

2 h2 2!

h4 -

4!

_ _8 2 2

...

0

...

0

.

8= h2n-2 ~~

(2s

-

h2n-4

2)! (272 - 4)! h2n-3

h2n-1

___

,(2n

-

~

l)! (2n - 3 ) !

...

h2 $2 - _ _ 2! 2

...

h3 3!

(4.2.21)

h

premultiply both sides of (4.2.20) by the adjoint B* of 8, considered as a matrix. Let 1 B 1 be the determinant of 8,and denote the elements of 8" by P z i . Then there results (4.2.22)

Thus we have (4.2.23)

I 9 IMj/h == P T n S j , (i = 1, 2 ,...)n - 1). 1 9 1Ai.j P,*,,& Let us now compare (4.2.23) with (4.2.5). We have 1

PL

82

=

=

n-1

(- T)

and so 2"-11 9 IMj

(4.2.24)

c d??)

h2n-1

2n--2

(2n - l)!

k-0

Mj--n+l+k

*

(4.2.25)

Iv.

130

THE POLYNOMIAL SPLINE

This identity can, of course, be established directly from the expansion of I 9 I. T h e derivation, however, is somewhat lengthy. T h e operator in the right-hand member of (4.2.24) may be written as 0

...

0

0

0

- -8 2 2

hZi-4

h26-2

______

(2i - 2 ) ! (2i - 4 ) ! h2d

h2t-2

-~ (2i)!

(22

-

2)!

h2i

h2i+Z

...

...

_ _ _ (2i 2)! (2i)!

+

h2n-4

...

h2 - -8 2 2! h4 4! h6 6!

2 h2 2! h4 -

0

...

0

0

...

0

_ - 12 2

4!

0

...

0

h2n-6

_______

0

. I .

(2n - 4)! (2% - 6 ) ! h2n-2

_

_

h2n-4

(2% - 2 ) ! (2n hZ 2! h4 __

4!

_

-

4)!

82

--

h2d-4

h2i

h2i-2

2

0

2 h2 2!

h2i-2

82 -_

...~

__

12

2

______ (2i - 2 ) ! (2i - 4)!

-~ (2i)! (2i - 2 ) !

...

...

...

0

...

0

h2 2 h4

4!

xi-,

i-

__

82

(4.2.26)

2 h2 2!

We note that, if we define 1 2! 1 -

4!

1 6! 1

-X

1 2! 1 4!

1

~ _ _ :2n - 2)! (2n - 4)! 1 1 _ _ _ _ _ (2n)! (2n - 2 ) !

0

0

...

0

-x

0

...

0

-X

...

0

1 2!

... _

...

1 2! I _

4!

-X

-1

2!

(4.2.27)

4.2.

131

EQUAL INTERVALS

then expanding in terms of minors of the first column gives

This is valid for n 3 2 if we define Q1 = 1/(2!). If we express Qn(x) as Qn(X) = an.0

+

an,lX

+ + *.*

(4.2.28)

a n , n - i Xn-l 9

then we have

Thus,

I n particular, =

1/(2!)n.

(4.2.30)

We see further that h2/2!= h2Q,(x),

and by induction that h2/2! h4/4! ha/6!

0

-X

h2/2! h4/4!

-X

h2/2!

... ... ...

0

0

0 0

0 0

=

hang,(%).

(4.2.31)

132

Iv.

THE POLYNOMIAL SPLINE

We turn now to the determination of the quantities Ak.jin the case of type I and type I1 splines. From (4.2.16), we have Spz-2P-1)

(Xj-1) =

-Mj-l

-

(2p)! ~

,& P

h2p-1

h2p-2k

( 2 p - 2k)! ' (4.2.32)

For type I splines, when the quantities M j have been found, we may determine Si*)(a)( q = 2n - 3, ..., N ) from (4.2.15) rewritten as

For type I1 splines, we similarly determine S$*'(a)(q = 1, 2, ..., n - 1). Once all the quantities Si*)(a)(q = 1,..., 2n - 3) have been found, we may determine the quantities Siq)(xj) by integration. Thus,

as may be seen by integrating by parts twice. 4.3. Existence

For polynomial splines on nonuniform meshes, existence proofs based directly upon the properties of the coefficient matrix are quite difficult to construct. We note that, even for the case of equal intervals,

4.3.

133

EXISTENCE

the coefficient matrix for the splines of degree 7 does not have dominant main diagonal (cf. coefficient table in Section 4.2). For polynomial splines of odd degree which are of best approximation or of minimum norm, the existence is a consequence of the first integral relation, as in the case of the cubic spline. These properties are explained in detail in Chapter V. I n the case of periodic polynomial splines on equal-interval meshes, nevertheless, existence may be handled directly. T h e proof leads, moreover, to several properties of the coefficient matrix which are of importance in themselves. T h e coefficient matrix in (4.2.1) is the circulant matrix of order N ,

A=

= AP-1) ,

a,

AT"-' = A?:-' ab = 0 otherwise.

(k

= aN+2--k =

=

1 , 2,..., n);

It is well known (cf. Aitken [1958] or Muir [1960]) that the eigenvalues of the matrix A in (4.3.1) are 6,

=

a,

+ azwk + .-.+ aNWc-'

(k

=

1, 2 ,..., N ) ,

(4.3.2)

where wk = eaakiIN.T h e corresponding eigenvectors are (1, w, , w ;

)...)u p ) .

T hus we may represent the matrix A as A

=

(4.3.3)

QnOQ-l,

where

(4.3.4)

(4.3.5)

Iv.

134

THE POLYNOMIAL SPLINE

We have

(4.3.6)

and, if none of the eigenvalues is equal to zero, A-1

= QO-lQ-1

=

C

N

N

N

k-1

k=l

(C 8i1/N, C (wk8J1/N ,..., k=l

The difference equation (4.2.6) has been studied by Hille [1962, p. 471 in respect to its association with the differential operator 0 = xd/dz in the complex plane. Let (4.3.8)

where the quantities +Lm) are as defined in (4.2.3). We show by induction that (4.3.9)

First,

Then, assuming

(4.3.10)

4.4. since q5Lm) = (k that we have PI(.) = Z,

+

135

CONVERGENCE

+ ( m - k)+fl14T1). From +

P,(z) = z [ ~ P , - ~ ( z ) (1 - z)PA-~(z)]

(4.3.10), we see ( m > 1).

(4.3.11)

We show also by induction that P,(z) has n - 1 distinct real negative zeros. I), so that - 1 is a zero of P2(z).Assume We note that P2(z)= Z(Z that P,-l(z) has m - 2 distinct negative real zeros in addition to vanishing at z = 0. Since Pm-l(x) is real (x = %z),there must be a zero of Pk-l(x) between two successive zeros of Pm-l(x). Thus, P,-l(x) and P,(x) do not simultaneously vanish except at x = 0. At a zero of Ph-l(x), P,(x) and P,-l(x) have opposite sign. P,(x) is monic so that, as x --t - co, P,(x) -+ - co or 03 according as m is odd or even. T hus P,(x) has a zero between each two consecutive zeros of PkPl(x) and a zero to the left of the left-most zero of Pm-l(x). I n addition, P,(O) = 0, and, since Ph(0) > 0, there is a zero of P,(x) between 0 and the right-most zero of Pk-l(x). Consequently, P,(x) has n - 1 distinct negative real roots in addition to the zero root. This completes the inductive proof. = as may be easily shown in a variety of ways. Now +im) Thus, the reciprocal of each zero of P , ( z ) / z is again a zero, and P2,( - I ) = 0. Thus, for periodic splines of odd degree, the eigenvalues Ok are all different from zero. [Indeed, they are all real, as may be seen , (K l N ) . ] T h e matrix A is therefrom (4.3.2), where a, = ~ ~ + ~ -< fore nonsingular in this case. We note that (-1) is an Nth root of unity only if N is even. T h u s for even-degree splines the matrix A is nonsingular if N is odd. This concludes the proof of the following theorem.

+

+

+r>l--k,

<

Theorem 4.3.1. Periodic polynomial splines of interpolation of odd degree N on a uniform mesh always exist. Periodic polynomial splines of even degree N exist if the number N of mesh intervals is odd.

<

<

4.4. Convergence A complete array of convergence properties analogous to those presented for cubic splines in Chapters I1 and 111 is not yet available for polynomial splines in general. If f(x) is of class @[a, b] (0 q 2n - 1) and SA,(x)is the spline of interpolation tof(x) on A , satisfylng certain

< 5

136

Iv.

THE POLYNOMIAL SPLINE

rather general end conditions and the sequence of meshes {A,} has lim,+w j j A , 11 = 0, we might expect that

f ‘ ” ’ ( ~ )- S$)(X) =

A , II”-”)

(0

< p < p < 2n - 1),

(4.4.1)

provided in the case of derivatives of order 2n - 1 that the quantities R A k = {maxi 11 A , I ~ / ( x , , ~ - x ~ , ~ - are ~ ) ]bounded. T h e authors have demonstrated convergence of sequences of periodic polynomial splines for asymptotically uniform meshes (Ahlberg et al. [1965]) for functions of class C2n--2[a,61. More recently, Ziegler [1965] has announced a proof of convergence for periodic splines without the restriction of asymptotic uniformity on the meshes. A recent result for generalized splines (Ahlberg et al. [1965a]) gives estimates of the order of convergence when f ( x ) is in Cn[a, b] and when f ( x ) is in CZn[a,b] provided in the latter case the meshes satisfy (2.3.7) (see Chapters V and VI) . I n Chapter V, we establish the fact that, if f ( x ) is in S n ( a , b ) , then

Furthermore, if the quantities RAkare bounded as k -+ co and if f ( x ) is in Z m ( a ,b), then f ‘ ” ’ ( ~ )- S$)(X)

=

O(1l A , i””-”-’)

(0

< p < 2n - 1).

As in the case of the cubic spline, the apparent defect in the rate of convergence arises from the method of proof employed. I n order to get at the more precise relation (4.4.1) as for cubics, it appears necessary to use properties of the inverse of the coefficient matrix for the equation system determining the spline. As of the present time, these properties have not been demonstrated except in certain special cases. I n this section, we prove some convergence results for general polynomial splines of odd degree which anticipate properties of the inverse coefficient matrix A for the spline equations described in Section 4.1. This is followed in the next section by a detailed discussion for periodic polynomial splines on uniform meshes.

Theorem 4.4.1,

I/ A , 11 -+ 0 as k

Let {A,} be a sequence of meshes on [a, b] with

+ co. Let f ( x ) be of class C2n-2[a, b]. Let Sdk be the

polynomial spline of degree 2n - 1 of interpolation to f ( x ) on A , such that f ( x ) - S A k ( xis) of type I’ or type II’ or such that SAk(x)is periodic [provided f ( x ) is periodic and of class C2n-2( - co, a)]. If the coejicient matrix A,

4.4.

137

CONVERGENCE

associated with the spline equations (4.1.5) and the end conditions has an inverse Akl which is bounded with respect to k , then uniformly on [ a , b] f'"'(X)

== O(l1

- s%'(X)

A , I ""-"-")

( p = 0, 1,..., 2n

If f 2 n - 2 ( x ) satisjies a Holder condition of order then f'"'(x) - SX'(x)

=

O((lA , l~zn-z+a-~)

(0

< 01

-

2).

(4.4.2)

< 1) on [ a , b ] ,

( p = 0, 1)...,2n

- 2).

(4.4.3)

uniformly with respect to x in [ a , b].

Let us drop the mesh index k , denote by M the vector MN)T, and let A represent the coefficient matrix of the (MO spline equation system. Write the spline equations as Proof. 1

I...)

AM

(4.4.4)

= d.

If S A ( x )is periodic, then (4.4.4) is precisely the system of equations (4.1.5) ( i = 1, 2,..., N ) . Here d is the vector ( d l ,..., d N ) with (4.4.5)

di = 8 y f ( x z ) . T hus we have A [ M - (2n - 2)! d]

=

[ I - (2n - 2)! A] d,

(4.4.6)

where I is the unit matrix of order N . In view of (4.1.7), the elements of the vector on the right may be rearranged as

T h e quantities Aij are nonnegative and consequently, in view of (4.1.7), are bounded uniformly with respect to the mesh index k. Moreover, = 0 for j < i - n + 1 and j > i n - 1. For i--n+ 1 i + n - 1,

+

<

1

dj

-

di

I

< p ( f ' 2 n - - 2 )(2n ; - 2)/1A 1 ).

(4.4.7)

Iv.

138

THE POLYNOMIAL SPLINE

Thus, we write

From this point, the proof for the periodic case proceeds in a familiar fashion. By a (2n - 3)-fold application of Rolle's theorem and the fact that S,(x) interpolates tof(x) at the points of A , we find that in any (2n - 2) consecutive intervals there is at least one point 5 at which Si2n-3) (5) = f ( 2 n - 3 ) (6). By writing - ~p-3)

f(2"-3)(x)

(414

we find that max lf'""-3'(x) la.bl

-

Sk-3'(x)j

< (28 - 3)11 A-l [I ~ ( f ( ~ " - '(2n );

-

2)1/d 11)

*

11 d 11.

Relations (4.4.2) and (4.4.3) now follow for the periodic case. Consider next the type I1 splines. Here the proof differs from that employed in the periodic case because of the n - 1 end conditions to be imposed at each end. Since we have explicitly assumed the uniform boundedness of I / A-l 11 (the existence of A-l is shown in Chapter V), we need only show that the right-hand member of (4.4.6) approaches zero as before. 1 . In addition We take Eqs. (4.1.5) for i = n - 1, n, ..., N - n to these we have the n - 1 relations consisting of

+

M0

and Eqs. (4.1.8) ( q

a

=

n, n

- f(2n-2)

+ 1,..., 2n

-

(4.4.9.1)

(0)

3) written in the form

{[O, x1 - a ,..., x, - a]2n-2}--1.

(4.4.9.2)

For a polynomial p ( x ) of degree not greater than 2n - 2, the - 2)!. right-hand member of (4.4.9.2) is precisely p(2n-2)(x0)/(2n Since (4.1.8) is an identity for polynomials of degree 2n - 2, set p ( x ) = ~ ~ " - ~ / ( 2 2)!. n Then each M i = 1 in (4.1.8). Also, we have represented the coefficient of M i in (4.1.8) as Aq-n+l,j[O, x1 - a ,..., xq Thus, E ~ = o A p - n + l ,j 1/(2n - 2)!. It remains to show that the right-hand member of (4.4.9.2) approaches

<

4.4.

139

CONVERGENCE

(x,,)/(2n - 2)! as I/ A Ij + 0. Let p ( x ) be the polynomial of degree 2n - 2 defined by ~ ( ~ ) ( x = , , )Sip)(xo),01 = 0, I, ..., 2n - 2. T h e n from pn-2)

it follows that

p [ x o )...,x,]

c k!1

2n-3 -

-&'(a)

[O, x1 - a,..., x,

*

-

a]

k

k=q

-

1

(2n - 2)!

...,x,

[O,

Sp'(a)

XI - a,

- a12n-2

and that the right-hand member of (4.4.9.2)may be written in the form

T h e first term in the last expression is the quotient of qth divided differences, over x,, , x1 ,..., xq , of a function F = S, - p whose first 2n - 2 derivatives vanish at x, and of the function X ~ R - ~ . Consider the more general problem of evaluating the quotient of divided differences of two K-fold differentiable functions, (4.4.11)

We note that we can write the kth divided difference in the form (cf. Davis [1963, p. 461)

F[xO 3

x 1

,...,x k ]

=

1 xo 1 Xi

xo2

..

.

...

x12

1

Xk

Xk2

.. .

."

... ...

2 - 1

F(xJ

X;-l

F(Xl)

x;-1

F(Xk)

,

.

1 1

...

1

xo x;

.'.

x1

...

x12

...

Xk

Xk2

."

.. .

x;

x :

. (4.4.12)

XkL

Consider then (cf. Goursat-Hedrick [1904, p. lo])

T h e function +(x) vanishes at x = x,, , x1 ,..., x ~ - If ~ .K is set equal to (4.4.1l), then +(xq) = 0 also. Th u s +'(x) vanishes K times in (xo, xp),

140

Iv.

1 #"'(X)

=

THE POLYNOMIAL SPLINE

xa

.1 XI . . . 5-1

Thus, for some (, xo < 5

"'

xk-1

"'

Xk-1

xa2 x12

2

5 - 1

...

.'

[F(')(x) - KG(')(x,)].

&l 5-1

< x k , we have

Hence the first member of (4.4.10) is equal to

the limit of which, as 11 A 11 + 0, is 0. I t follows from (4.4.10) now that the limit of the right-hand member of (4.4.9b) as 11 d /I + 0 is Si2n-2)(x,)/(2n- 2 ) ! T h e remainder of the proof for type I1 splines proceeds as in the previous case. T h e proof for type I splines is omitted. No new ideas are introduced, and the proof is largely involved with the elimination of the quantities Sl;")(a),SLn+l)a,..., Si2n-3)(a)from Eqs. (4.1.8)-a straightforward but somewhat tedious process. We point out certain important cases in which the uniform boundedness of 11 A-l 11 has been established and to which the theorem applies. T h e authors have shown (Ahlberg et al. [1965, Theorem 41) that this is the case for periodic splines on a sequence of asymptotically uniform meshes (A,): II 4 II 0 and [maxi I - hk,i+l/(hk,i hk,i+l)ll 0. I n particular, periodic splines on uniform meshes will be considered subsequently. A second general result may be demonstrated for the case in which continuity requirements on the approximated function are relaxed but the additional mesh requirement (2.3.7) imposed.

-

4

-

+

Let { A k } be a sequence of meshes on [a, b] with and RA, = [maxj I1 A , ll/(xk,j+l - xk,j)] B < 00. Let f ( x ) be of class Cg[a, b] (0 q 2n - 3 ) . Let the norms 11 Azl 11 of the inverses of the coeficient matrices be bounded as k -+ m. Then f o r the polynomial splines SA,(x) of degree 2n - 1 of interpolation to f ( x ) on A ,

Theorem 4.4.2. 11 I/ -+ 0 as k -+

< <

<

4.4.

141

CONVERGENCE

such that f ( x ) - Sdk(x)is of type 1', type 11', or such that SAk(x)is periodic [;f f ( x ) is periodic and of class C*(- 00, a)], we have

If f(")(X)

( p = 0, I, ..., 4 ) .

A , 1 "-)

[f'"'(x) - S ~ ' ( X )= ]

satisjies a Holder condition of order a(0

< 01

(4.4.13)

< 1) on [a, b ] , then

( p = 0, I , ..., p).

[ f ( " ' ( x )- S ~ ' ( X ) = ] O(ll d, il"+"-')

(4.4.14)

Proof. We limit our arguments to the cases of periodic splines and type I1 splines, as in the case of the preceding proof. For q < 2n - 2, the divided difference (dropping the mesh index) f[xj-n+l ,..., xi+n-l] satisfies

I f [ ~ j - n + l > - * .~>f t n - 1 1 I

22"-n-2

< (2n

-

,-,-,

2n q 2

2 ) !II

-

II

C L [ f ( " ) ; (4

+ 1)Il

Ill. 14.4.15)

T h e right-hand members of (4.4.9.2) can be shown to satisfy a similar inequality involving a constant multiple of I l [ P ; (4

+ 1)Il

1ll1Il 1 2n-q-2.

It then follows from the uniform boundedness of 11 A-l 11 that the spline equations (4.1.5) augmented by (4.4.9), etc., give, for some KO independent of the mesh index, the inequality [max II

112n--9-21

M f11 < KotL[f(q';( 4

+ 1)ll

Ill

< Ko(4 + 1) tLCL(f(a'; II

11).

(4.4.16)

Hence, [mcx I/

1 2n-a- 21

SF-') (4 I1 < Ko(p + 1) C L ( f ( " ) ;

I1 1 ).

From (4.1.13), it may be concluded in the periodic case for K p independent of the mesh index that

this inequality is valid as well in the nonperiodic case when n-1< j N - n + 1, and a similar inequality results for 0 < j < n - 1, N -n 1 <j N if we employ the boundary conditions. It now follows from (4.1.11) that for modified constants K $

+

<

<

II 1 2n-"-a- 2

I Skb) I

< K;Il(ffa);II

II)

Iv.

142

THE POLYNOMIAL SPLINE

and hence that

Also Sd(xj) = f ( x j ) , and we havef[xi-q ,..., xi] = SZp)(tl) Thus, on [xi-l, 9 1 , some t1and tzin (xi-q+l , [SY'(x) - f ' " ( x ) ]

=

[Sk'(x) - S$)(X$)]

+ [SY)(Xj)

+ [ f ' " ( e z ) -f'"(x)l, I < [~,*,,(l+ s) + sl P ( f

-

=

f(@(tz) for

S$'(&)]

I Sk'<.) -f"'(x) (% /I A 1 ). Relations (4.4.13)and (4.4.14)are now a direct consequence for

= q. T h e relations for p < q follow again by integration and the use of the interpolation property. I t may be seen that the proofs of the preceding theorems rest upon certain boundedness properties of the quantities M irather than directly upon the boundedness of the inverse matrix norms ( 1 /I. Let f ( x ) be b]. If I Mk,i 1 B ( j = 0, 1 ,..., Nk ; k = 1, 2,...) and of class C2n--2[a, if the sequence of meshes is nested, it may be shown for the polynomial splines of interpolation S,*(x) that {Si?-')(x)] converges uniformly to f ( 2 n - 2 ) ( xon ) [a, b] (see Ahlberg et al. [1965], Theorem 3). On the other hand, we can remove the requirement of being nested if we assume that the S L:-')(x are equicontinuous. For, by the interpolation property, Sir-')(x) = f(2n-2)(x) at least once in any 2n - 2 consecutive intervals, and so

p

<

1

-f ( a n 4

(Xi

I

< P[f'2n-z'; .(

- 1)ll A ,

Ill

+ P [ S p ;.(

- 1)ll A ,

Ill.

We may further show (see Ahlberg et al. [1965], p.237) that, if the meshes are nested and { S d k ( x ) }is a sequence of splines interpolating to an arbitrary function f ( x ) defined on [a, b] with SiF-')(x) equicontinuous and bounded at one point, thenf(2n-z)(x)exists, and ST-"(x)-+f ( 2 n - 2 ) ( ~ ) uniformly on [a, b]. Theorem 4.4.2may be modified as follows.

Theorem 4.4.3, Let {A,} be a sequence of meshes on [a, b] with as k + co. Let f ( x ) be of class O [ a , b] (0 < q < 2n - 3). Assume for the polynomial splines Sdk(x)of interpolation to f ( x ) on A , that

Ij A , 11 -+0

l i ~ ~ { m a ~M,,j j [ j 1 Ij A ,

k+x

Il"-"-"])

0.

4.5.

QUINTIC SPLINES OF DEFICIENCY

2, 3

143

Then on [a, b] [f‘”’(x) - sk’(x)l

=

.(ti A , II“-”)

(0

< P < q).

Proof. T h e proof is merely an obvious modification of the proof of Theorem 4.4.2 beginning at (4.4.16).

4.5. Quintic Splines of Deficiency 2, 3 T h e first integral relation developed in Chapter I11 for cubic splines is extended in Chapter V to include splines of odd degree. As we shall see there the condition for the validity of this relation for a function f(x) of class Cn[a,b] and a polynomial spline of degree 2n - 1 is that

c 2 [f‘”-”’(x) N

n

j=1

”=I

-

sy-”(x)] Sy+v-l’(x) IX9

= 0.

Xl-1

We pass over, for the time being, the spline end conditions that are relevant to this constraint and observe that for a simple spline (i.e., of class C2n-2[a,b ] )the terms f o r j = 1,..., N - 1, p = 1, 2,..., n - 1 drop out because of the continuity of SJn+p-l)(x) at xj . T h e terms involving S:n-l(xj) vanish because f(q)- S(xj) = 0. We discuss in Chapter V the fact that it is possible to replace the requirement of continuity on SJn+p-l)(x)by the condition

s p ( x j )=f h-”’(xi)

< < <

for all p satisfying 1 q p n - 1 ( q prescribed). I n the case of the quintic spline, the maximum deficiency permitted is 3 ( q = 1). Here

sy(xi) =fy’(xj)

( p = 0, 1, 2; j

=

1 ,..., N - 1).

I n setting up procedures for approximating a given analytic function in numerical integration, differentiation, or interpolation, it is frequently practical to use such higher-order interpolation. This is an important application of splines with deficiency greater than 1. A different application of this concept which is of considerable interest, however, involves the use of splines of maximum deficiency when only the function f ( x ) itself is known or can conveniently be determined at mesh points. Here we first approximate the derivatives required by some auxiliary device, such as by using polynomials of interpolation. *

* The effectiveness of this quintic spline in curve-fitting together with its relation to Simpson’s rule and the three-eighth’s rule was pointed out by S. Auslender.

Iv.

144

THE POLYNOMIAL SPLINE

For quintic splines of deficiency 3, we take the approximations to andfJ!‘ from parabolas of interpolation. For 0 < j < N , let uj = ui(x) represent the parabola passing through (xjP1, &), (xj , f i ) , and (Xi+l 9 fi+l). Thus,

fj‘

u,(x,)

=f, ,

Ui.(X,)

= A,

U;(xj) =

f, - L 1

+ ~ L f,+l j

h,

-f,

h,+l

2f[x,-, , xj I

(4.5.1)

’

, X,+J.

+

Here, as in Chapter 11, Aj = 1 - pi = hj+J(hi hi+l),. T h e quintic spline $(x) of deficiency 3 is defined over the interval [xi-1 , 5 1 by

S$’’(X+*) = U~?\(X,-~),

S?’(X,)

( p = 0,1, 2). (4.5.2)

= U;”’(X,)

Now we have the equations Uj(.)

Let a(.) Then

= f,

+ (x

-

x,) U p , )

uj(xj-l)

= f, - h,Ui(X,)

Uj(Xj+,)

= f,

+ (x

-

xj)2u;(xj)/2!,

+ hj2U;(Xj)/2,

+ h,+,.;(.,,

(4.5.3)

+ y+,u;(x,)/2.

be the cubic satisfying a(0) = a’(0) = a’(1) =

32

-

=

0, a(1)

-

.,(.)I

+ %(.)

1.

(4.5.4)

2u3,

and we may express the quintic with which $(x) coincides on as ”.(I-,.

=

44,

xi]

(4.5.5)

where aj(x) = a[(x - ~ , - ~ ) / h ~ ] . We note at this juncture that the first and second derivatives at xi are weighted means of the slopes mi-l , m i ,mitl and moments Mip1, Mj, of the simple cubic spline [see (2.1.15) and (2.1.7)]. For equal intervals this represents a smoothing, in the sense of Schoenberg, of the first and second derivatives of the cubic spline. I t has been found in practice that there is occasionally an advantage in curve-fitting in using this quintic spline instead of the simple cubic spline. It is also important to observe that the quintic spline of deficiency 3 is locally completely determined from the approximated function and its first and second divided differences. Thus, no simultaneous system of

4.5.

Q U I N T I C SPLINES OF DEFICIENCY

2, 3

145

equations need be solved for the determination of these quintics. I n addition, four parameters suffice to describe the quintic on [ x , - ~ , x,]. We have &(x)

= {f?-1

+ (x

-

x,-1) L ( x , - 1 )

+ (x

+ if, + (x - X,).;(x,) + (x

-

-

~ , - l ~ " u ; ' _ l ~ ~ , - l ~/ ~ 441 ~>[1

x3)24(x,)/2!) +9.

(4.5.6)

Since uXx3) =

u;&-l)

=

fj

-fj-1

-

hi

hi &xj-l),

2

we have $(x)

xj - x = fjp1-

-

hi

(Xj -

x +fj

x)(x

-

-

xj-1)

2

xj-1

hj

{u;-l(xf-l)[l

- a&)]

+ u;(xj)

a&)}.

(4.5.7)

There are significant relations of this type of spline to numerical integration. We obtain hi

fi-1 2+fi

hi3 24

[4-1(Xj-l)

+

4(Xj)I,

noting that we have employed f j - 2 ,fi-l ,f i , fi+l in this approximation. Let us set xi - xi-1 = h, - xi-2 = xi+l - xi = Ah. Then the approximation becomes

If we take h = - +, the effect is to integrate over three intervals with x j with common the abscissas arranged in the order x i - 1 , xi-2, interval of spacing H = h/3. T h e result is

which is the three-eighth's rule.

Iv.

146

THE POLYNOMIAL SPLINE

If we take X = - 8, the effect is to integrate over two intervals with abscissas arranged as xi-1, xiPz = xi with interval H = h / 2 . Here we obtain

(W)( fj-l

+ 4fjAl +fi),

which is Simpson’s rule. Suppose we choose X in such a way that the coefficients of fiPla n d f j vanish. We obtain 6h2

+ 6h + 1

h = (-3 & 3lI2)/6.

= 0,

T hus we have

which is Gaussian integration of second order. As may be expected, the convergence is effectively comparable to that of the cubic spline. On [xi-l,xi] we have from (4.5.7)

I S,(x)

-f”(x)

1

=

1 [u;-l(xj-l)(l

+ loo( 1

-

- o)

+

u;(.j)

cr

-f”(X)]

o)(1 - 20)[u;-1(xj-l)

G p ( f ” ; II 11) . [2

+ 116(3)1121.

-

u;(xj)] 1 (4.5.10)

If f ( x ) is of class C’[a, b], we have on [xi-l, xi]

(4.5.1 1)

4.5.

QUINTIC SPLINES OF DEFICIENCY

2, 3

147

Finally, if f(x) is of class C[a, b], then

and, if again 0

< /3-' < hj/hi+, < /3 for all meshes A , we have I h%;-I(xj-l) I


II 11).

Thus, it follows that

I sA(x) -f(.)

1

< const

'

p(f; 11

11).

(4.5.12)

I n another direction, we consider the extension to quintic splines of the ideas of FejCr introduced in connection with the proof of Theorem 2.3.1. Let us study the quintic spline of deficiency 2 which interpolates to a functionf(x) of class C[a, b] at the points of a mesh A on [a, b] and has zero derivative at each of these points. We confine our attention here to periodic quintic splines with f(x) in C(- CQ, 03) and of period b - a. Again designate this spline by sd(x), and, for convenience, set Ni= $(xi). From (4.1.22), we have the relations

(4.5.13)

continuity of sy(x) at xi gives the condition

148

Iv.

THE POLYNOMIAL SPLINE

Here the norm of the inverse of the coefficient matrix is bounded independently of d. If f ( x ) is continuous, then 1 N j I 11 d 11' $20f12(2fl- l)p(f; I/ A 11) if [maxjll d jl/hj] fl. From (4.6.13), it may be concluded that the quantities s:(xj) 11 d /I3 have similar bounds independent of d, and from (4.1.27) that the same is true of si;"(xj -)I( d , xi] we have and SJ4)(x,+) / / d (I4. Since on

<

<

we may conclude that SA(x) -f(x) may be made arbitrarily small by taking 11 d 11 sufficiently small, provided (2.3.7) is satisfied. We have thus proved the following theorem.

Theorem 4.5.1. Let {A,} be a sequence of meshes on [a, b] with -+ 00 and satisfying (2.3.7). Let f ( x ) be of class C( - 00,CO) and of period b - a . Let SAk(x)be the periodic quintic spline of deJiciency 2 interpolating to f ( x ) and with Si,(x) = 0 at the points of A , . Then {Sdk(x)} converges uniformly to f(x) as k -+ 00.

I/ d, 11 -+ 0 as K

4.6. Convergence of Periodic Splines on Uniform Meshes T h e coefficient matrix in (4.2.5), which after multiplication by (2n - I)! becomes the circulant of order N ,

has been shown in Section 4.3 to be nonsingular. We want to show also that the norms of its inverse, for fixed n, are bounded with respect t o N . I t was established in Section 4.3 that the function (4.6.2) i-0

4.6.

CONVERGENCE OF PERIODIC SPLINES ON UNIFORM MESHES

149

has only real negative roots different from -1. T h e eigenvalues of (4.6.1) are the numbers q(wk)/wE-', where w k = exp(2kri/N).Thus the eigenvalues are bounded away from zero. T h e following theorem on circulants now gives us the property needed.

Theorem 4.6.1. Let ck be a sequence of circulants c(aik),uik), ..., a e ) , 1, 2, ..., . Let aj ( j = -m 2, -m 3,..., m ) be given, real or complex, and suppose that a:,) = aj ( j = 1,..., m), a!,) = U j - N , ( j = Nk - m 2, ..., Nk), a!,) = 0 otherwise. If the polynomial

k

+

=

+

+

q(x) = Cjm=-m+2ajzj+m-2 has no roots of modulus unity, then Czl exists, and 11 Cil I/ is bounded as k -+ 00. Proof.

Let Pk be the permutation matrix of order Nk : Pk

Then Pk

=

C(alk),agk),...,a$)

c(0,1, o,...j0 , 0). =

c(a:

, a:,) ,..., uNk-l ),( ). Thus

cI,= up)^^ + a P P k + + a$:Pp-1 = P,"+'q(P,),

where I k is the unit matrix of order N k . Since the polynomial q(z) has no zeros of modulus unity, there exists an annulus containing 1 x 1 = 1, rl < I x I < r 2 , in which q(z) # 0. I n this annulus, [X-"+'~(X)]-~ may be expanded in a power series CZ-,.bjzi which converges absolutely. Since I/ Pk I/ = 1 (row-max norm), the series Cj"_-,bjPkj converges in this norm to the inverse Ck1 of C , . Furthermore,

I1 GII <

c m

j=-a,

I bj

I1

where the right-hand member depends only upon a-m+2 ,..., a,. T h e following theorem is now an immediate consequence of Theorem 4.4.1.

Theorem 4.6.2. Let f ( x ) be of class C2n-2(- CO, a)and have period b - a. Let {dk} be a sequence of uniform meshes on [a, b] with limk+* 11 dk 11 = 0. For a given n > 1 , let S d k ( x )be the periodic polynomial spline of degree 2n - 1 on A, interpolating to f ( x ) at the mesh points. Then [f'"'(X)

- Sk'(X)] = o(l1 A ,

112n--2-")

( p = 0, 1 , 2,*..,2n

-2)

Iv.

150

THE POLYNOMIAL SPLINE

uniformly with respect to x in [a, b]. If f2n-2(x) satisfies a Holder condition of order a on [a, b] (0 < a I), then

<

[ f ‘ ” ’ ( x )- sk)(x)]= o(ild k l12n-2+o- p

(p

)

= 0 , 1,

..., 2n - 2)

uniformly with respect to x in [a, b]. We consider next the strengthening of the differentiability properties Off(4.

Theorem 4.6.3. Let f(x) be of class C2n-1(-co, co) and have period b - a. Let {A,) be a sequence of uniform meshes on [a, b] with limk-,m 11 A , 11 = 0. For agiven n > 1, let SAk(x)be the periodicpolynomial spline of degree 2n - 1 on A , interpolating to f(x) at the mesh points. Then, uniformly on [a, b],

where SJF-’)(xj) may be taken either as the right-hand or left-hand limit at the mesh point xj . Iff (2n-1)(x)satisfies a Holder condition of order a on [a, b] (0 < a l), then

<

[ f ‘ ” ’ ( x ) - sk’(x)] = O(1l d k 1)2n--l+ol-” )

( p = 0, 1,...,2n - 1). (4.6.4)

uniformly with respect to x in [a, b]. Proof.

We set (dropping mesh index k )

and obtain from (4.2.5) by subtracting members of each equation from corresponding members of its successor

Thus, if C is the circulant (4.6.1), u the vector (crl ,..., oNk),and r the vector representing the right-hand members of (4.6.5), we have 1

(2n - l)!

C.(u-r)=

(I -

(2n - I)!

C ) r.

(4.6.6)

4.6.

CONVERGENCE OF PERIODIC SPLINES ON UNIFORM MESHES

151

Here the right-hand member may be put into the form

where we have used r+ = r N k + , r N k + j = r. T h e norm of the right-hand vector of (4.6.6) is, therefore, not greater than (3n - 3)~(f(~"-l); 11 A , 11). Thus,

11 u - T 11

< (2n - I)! I] C-I Il(3n - 3) p(f'2"-"; I/ A , 11).

On the other hand, on [xi-l,xi],

If'2n-"(~)

- T,

On [xi-l,xi], SJ2"-l)(x) [ sup 5

On [ a . b ]

I sy-1yx) -f(2 + 1 )

1

< p(f('"-''; (2n - l)li A , 11) < (2n - l ) p(f'2n-1';I/ 11).

= oi

(x)

. Thus, I]

< [(2n - 1) + (2n - I)! I1 C-'

Il(3n - 3)1

. p ( f ( 2 n - 1 ) . I1 A , 1 ). 9

This establishes (4.6.3) and (4.6.4) for p = 2n - 1 . B y integration and the interpolation property of SA(x),these relationships may be established for the smaller values of p in the usual manner (cf. Theorem 2.3.3). Similar methods yield the extension of Theorem 2.9.5. We merely state the result.

Theorem 4.6.4. Let f ( x ) be of class C2n(- co, co) and of period b - a. Let {Ak} be a sequence of uniform meshes on [a, b] with limk-tm/I dk /I = 0. If Sdk(x)is the periodic polynomial spline of degree 2n - 1 of interpolation to f ( x ) on A , , then k+m lim

[ma,

I

st-')(%k,j

+) - st-''(%,,j -) f(2n)(Xk,j) hk,j

-

2

I]

= 0.

152

Iv.

THE POLYSOMIAL SPLINE

Other convergence results relate to the weakening of the requirements on f ( x ) . We state these in the following theorem, which follows in consequence of Theorem 4.4.2.

< <

Theorem 4.6.5. Let f ( x ) be of class Cq(- co; a)(0 q 2n - 2 ) and of period b - a. Let {Ak) be a sequence of uniform meshes on [a, b] with limk+* / / A , 11 = 0. Let S,(x) be the periodic polynomial spline of degree 2n - 1 of interpolation to f ( x ) on d k . Then, uniformly on [a, b],

[f‘”’(.)

-

(0 < P

s 3 4 = o(ll A , 1 “”) !

< 4).

I f f ( @ ( x satisfis ) a Holder condition of order a on [a,b] (0 < a [jy.)

s~(= ~ o(ll ) lA , p+a--p

uniformly with respect to x in [a, b].

)

(0 < P

< 4).

(4.6.7)

< l), then (4.64

CHAPTER V

Intrinsic Properties of Polynomial Splines of Odd Degree

5.1. Introduction

T o a large extent, this chapter parallels Chapter 111, and almost every theorem obtained for cubic splines in Chapter I11 has its analog here. T h e major change that occurs is that, when considering a polynomial spline of degree 2n - 1 ( n > l), we replace the function space Z 2 ( u ,b) by the function space X " ( a , b) and the inner product

by the inner product

Chapter IV already has brought to light some of the problems that occur with splines of even degree and has indicated several ways of circumventing part of these difficulties. I n this chapter, the requirement that the splines under consideration be of odd degree is of paramount importance, and the requirement cannot be relaxed. T h e development of the intrinsic properties for generalized splines, which include polynomial splines of odd degree as a special case, again requires the introduction of an inner product (f,g) into Z n ( u , b). This can be done when the splines satisfy, except at mesh points, a self adjoint differential equation AS,

=0

(5.1.3)

of order 2n. I n this case, A has a factorization A =L*L 153

(5.1.4)

v. POLYNOMIAL

154

SPLINES OF O D D DEGREE

where L is a linear operator of order n and L* is its formal adjoint. We then are able to define (f,g) as (5.1.5)

In Chapter VI, we adopt this very general point of view. I t is necessary, however, that L have real coefficients and no singularities. * When a proof in this chapter essentially repeats the proof for the cubic case given in Chapter 111, we only state the theorem. I n general, we limit the discussion to the differences in the argument. T h e uniqueness and existence theorems that we obtain in particular require some further elaboration. 5.2, The Fundamental Identity

Let Sd(x) be a polynomial spline of degree 2n - 1 on a mesh A : < x1 < < xN = b, and let f ( x ) be in X n ( a , b). We have the identity a = xo

Jb

a

{f'"'(x) - S?)(x)}2 ax =

sb a

{ f ' " ' ( ~ )dx } ~- 2

s

b

a

{ f ' " ' ( x ) - SF)(x)}* Sy)(x) dx -

sb 0,

( S ~ ' ( Xdx )}~ (5.2.1)

in which the second integral of the right-hand member may be written as b a

{f'"'(x) - Sp)(x)} S p ) ( x )dx

c jxi {f'"'(x) N

=

j-1

- SY)(x)}

. SF)(%) ax.

Ij-1

I n each mesh interval [xi-l, xj] we can integrate by parts n times; thus

* A similar theory for complex-valued splines on [a,b] is possible. The definitions of (f,g) and L*, however, require slight modifications.

5.3.

T H E FIRST INTEGRAL RELATION

155

It follows that

I n obtaining (5.2.2), we have used the continuity of {f(n-=)(x)

- p-"'(x)} S h + n - l ) (x )

on

[a,b]

f o r a = 1 , 2 ,..., n - I

but have imposed no requirement other than that Sd(x) is a simple polynomial spline of degree 2n - 1 on d and that f ( x ) is in X n ( a , b). Equation (5.2.2) is the fundamental identity for a simple polynomial spline of degree 2n - 1 on a mesh A .

5.3. The First Integral Relation It follows immediately from the fundamental identity that, if Sd(f;x) is a spline of interpolation to f ( x ) on A satisfying any of a variety of end conditions, we have the relation

which constitutes the jirst integral relation for a polynomial spline of degree 2n - 1 interpolating to a functionf(x) in %%(a, b) on a mesh d. We have the following theorem.

Theorem 5.3.1. Let d: a = xo < x1 < -.. < x, = b a n d f ( x ) in X n ( a , 6 ) be given. If Sd(f;x) is a spline of interpolation to f ( x ) on A and one of the conditions, (a) f ( x ) - S d ( f ;x) is of type I', (b) Sd(f;x) is of type II', (c) Sd(f;x) and f ( x ) are periodic, is satisfied, then

Clearly, we also can employ end conditions of mixed type; we can as indicated in Chapter I11 even relax the continuity requirements

v.

156

POLYNOMIAL SPLINES OF O D D DEGREE

imposed on S d ( f ; x ) and its derivatives and still obtain the first ' integral relation by requiring that certain derivatives of Sd(f;x) interpolate to corresponding derivatives of f ( x ) on A . These extra interpolation requirements replace the continuity of { f ( + a ) ( x ) - SL"-=)(f;x)} . SJn+=-')(f; x) on [a, b]. T h e following theorem expresses this possibility.

< xN = b and f ( x ) in Theorem 53.2. Let d: a = x, < x1 < X n ( a , b) be given. Let Sd(x) be in X z n - k ( a , b) (k n) and SL2")(x) = 0 on each open mesh interval of A. If Sy)(x) interpolates on d to f f m ) ( x > ( a = 0, 1,..., k - 1 ; when a # 0, x # a, x # b), then we have

<

s" a

{ f ( n ) ( x ) } 2dx =

+

{ S ~ ) ( X )dx} ~ a

i"

{ f l n ) ( x ) - S ~ ) ( X dx )}~

a

provided that one of the conditions, (a) f ( x ) - S,(x) is of type 1', (b) S,(x) is of type 11', (c) Sd(x) a n d f ( x ) are periodic, is satisjied.

REMARK5.3.1. In the periodic case this theorem, as well as a number of other theorems in this chapter, requires that Sd(f; a ) = f ( a ) and S y ) ( f ;a ) = S g ) ( f ;b) ( a = 0, I, ..., 2n - 2). T h e conditions S y ) ( f ;a ) = f ( " ) ( a )( a = 0, I , ...,k - 1) and S y ' ( f ;a ) = S F ' ( f ; b )( a =0,1,...,2n - k - I ) can also be used. 5.4. The Minimum Norm Property T h e establishment of the first integral relation for polynomial splines of degree 2n - 1 has as an immediate consequence the analog of the minimum norm property of cubic splines. I n Theorem 5.4.1, which follows, we express this property for three principal sets of conditions under which it is valid.

Theorem 5.4,l. Let d: a = x, < x1 < < x N = b and f ( x ) be given, where f ( x ) is an arbitrary function on [a, b] having n - 1 derivatives at both x = a and x = b. Then, of all functions g(x) in X n ( a , b) which interpolate to f ( x ) on A , the ty2e 11' spline Sd(f;x) minimizes

1;

{g(n)(x)}zdx.

(5.4.1)

If the functions g ( x ) are restricted so that f ( x ) - g ( x ) is of type 1', or ;f f ( x ) and g(x) are required to be periodic functions, then (5.4.1) is minimized by the spline of interpozation to f ( x ) on A sutisfying the same restriction.

5.5.

THE BEST APPROXIMATION PROPERTY

157

As in Theorem 5.3.2, we can relax the continuity requirements imposed on Sd(f;x). I n order to facilitate the discussion, we introduce the following terminology. Let a mesh A : a = xo < x1 < . * . < xN = b be given, and let Sd(x) be a function in X z n - k ( a , b) with the property that SJ2")(x)vanishes identically in each open mesh interval of d. If SA(x) satisfies these conditions, we call SA(x) a polynomial spline of k n. degree 2n - 1 with deficiency k. We impose the restriction 0 I n this terminology, a simple spline on A has deficiency one, and a polynomial of degree 2n - 1 has deficiency zero.

< <

Theorem 5.4.2. Let A : a = xo < x1 < ... < xN = b and f(x) begiven, where f ( x ) is an arbitrary function on [a, b] having n - 1 derivatives at x = a and x = b and k - 1 (k n) derivatives at the interior mesh points of A . Then, of all functions g ( x ) in Z n ( a , b) interpolating to f(x) on d and with g(')(x) interpolating to f(')(x)(a = 1, 2, ..., k - 1) at the interior mesh points of A , the type II' spline of deficiency k minimizes (5.4.1). If the functions g ( x ) are further restrictedso that f(x) - g ( x ) is of type I' or that f(x) and g ( x ) are required to be periodic functions, then (5.4.1) is minimized by the spline of interpolation of deficiency k satisfying the same restrictions.

<

Let g ( x ) be a function satisfying the conditions of either Theorem 5.4.1 or Theorem 5.4.2, and let g ( x ) minimize (5.4.1). Then g(x) must differ from the corresponding spline of interpolation &(f; x) by a solution of Dnf = 0; this follows directly from the first integral relation. In addition, g ( x ) - Sd(f;x) must vanish on A , and, depending on the auxiliary conditions imposed, at certain points of A some of its derivatives must also vanish. If the number of these interpolation requirements is n or greater, then we can expect that g ( x ) = S,(f; x). When, in particular, A contains at least n points, this is true. Moreover, if g ( x ) is required t o be periodic, then g ( x ) - S,(f; x) is periodic and consequently a constant; but the constant is zero, since g ( a ) = S,(f; a). I n Section 5.6, we examine this question in more detail.

5.5. The Best Approximation Property A polynomial spline Sd(x) on a mesh A depends linearly on its values at mesh points and linearly on its prescribed derivatives. Moreover, S,(x) is completely determined by these quantities. These statements hold whether or not the deficiency of the spline is one. I n particular, Eqs. (4.1.5), (4.1.8), and (4.1.1 1)-(4.1.13) of Chapter IV and Eqs.

v. POLYNOMIAL

158

SPLINES OF ODD DEGREE

(5.7.3), (5.7.4), and (5.12.3) of this chapter exhibit this linear dependence. Consequently, we have decompositions of the form SA(f

+ g;).

=

&(f; ).

+ SA(& x),

(5.5.1)

subject to appropriate selection of end conditions. These decompositions extend those obtained earlier for cubic splines, and here, as in Chapter 111, the most useful of these decompositions occur when one of the following conditions is satisfied: (4 all of (b) (c)

+

+

S,(f g; 4 - (f g)(x), S,(f; 4 - f(4,and S,(g; ). - g ( 4 are type 1'. S,(f g ; x) and S,(f; x) S,(g; x) are both of type 11'. S,(f g; x), S A ( f ;x), and S,(g; x) are all periodic splines.

+ +

+

One of the most important consequences of the decomposition (5.5.1) is that, ifg(x) is identical with a spline -S,(x),then (5.5.1) becomes Sd(f

- S A ; x) = Sd(f; x) - S&)

(5.5.2)

when any of the conditions (a), (b), or (c) is met.* Again (5.5.2) is valid under these conditions even when the deficiency of S A ( f - 8, ; x), S,(f; x), and S,(x) is k > 1. If we determine Sd(f- S, ; x) such that the first integral relation holds, it follows, just as in Chapter 111, that

(5.5.3) J U

T h e two theorems that follow are immediate from (5.5.3).

< xN = 6 and f(x) in Theorem 5.5.1. Let A : a = xo < x1 < &'-%(a, b) be given. Let S,(x) be a spline on A , and let S A ( f ;x) be the spline of interpolation to f(x) on A such that f(x) - S,(f; x) is of type 1'. Then

* Both S,(f;x) and -S,(g; x) are assumed to be in the same type I1 equivalence class as Sd(x) when condition (b) pertains.

5.6.

159

UNIQUENESS

If S,(X) is restricted to a prescribed type 11 equivalence class, the integral

s:

(5.5.5)

( f ( " ) ( x )- S ~ ) ( X dx) } ~

is minimized by the spline of interpolation tof ( x ) on A in the same equivalence class; if f ( x ) and S,(X) are required to be periodic functions, (5.5.5) again is minimized by the spline of interpolation to f ( x ) on A .

Theorem 5.5.2. Let A : a = x, < x1 < < xN = b and f ( x ) in Y n ( a , 6 ) be given. Let SA(x)be a spline on A of dejiciency k (k n), and let S,( f ; x ) be the spline of interpolation to f ( x ) on A of deficiency k such that f ( x ) - s A ( f ; x ) is of type I' and such that sy)(f; x ) interpolates to f ( " ) ( x )( a = 1,2, ..., k - 1) at the interior mesh points of A. Then (5.5.4) holds. Moreover, ;f SA(x)is restricted to a prescribed type 11 equivalence class, or i j f ( x ) and S,(X)are restricted to periodic functions, then (5.5.5) is minimized by the corresponding splines s A ( f ; x ) of dejiciency k, which interpolate to f ( x ) on A and have derivatives S y ) ( f ;x ) which interpolate to f ( = ) ( x (01 ) = 1, 2 ,..., k - 1) at interior mesh points of A.

<

If S,(X) also minimizes (5.5.5) and otherwise satisfies the requirements imposed on s , ( ~in) Theorem 5.5.1 or Theorem 5.5.2, then S,(X) - S A ( f ; x) satisfies Dny = 0. I n the periodic case, S,(X) - s A ( f ; X) is a constant.

5.6. Uniqueness T h e question of the uniqueness of a polynomial spline of degree < xN = b is easily reduced 2n - 1 on a mesh A : a = x,, < x1 < to the question of whether or not a polynomial P ( x ) of degree n - 1 which vanishes on A and has certain other properties vanishes identically. If P(x) is of type I' or if P(x) is periodic, P ( x ) vanishes identically. I n these two instances, A may consist of just the two points x = a and x = b. When P ( x ) is of type 11', the situation is different, since requiring that derivatives of order 01 ( a = n, n + 1,..., 2n - 2 ) vanish at x = u and x = b imposes no additional constraint on P(x). I n this case, A must contain at least n points to ensure that P ( x ) vanishes identically whenever it vanishes on A . If, however, P(u)(x)( a = 0, 1,..., K - 1) is required to vanish at the interior mesh points of A , then P ( x ) vanishes identically when k(N - 1) 3 n - 2 and P ( a ) = P(b) = 0. We observe that the zero function z ( x ) is a polynomial spline of degree 2n - 1 for any integral value of n greater than zero. Not only is z ( x ) of type 1', but it is of type 11' and is periodic as well. Moreover, z ( x ) can be interpreted as a spline of degree 2n - 1 on d of deficiency K

160

v.

POLYNOMIAL SPLINES OF ODD DEGREE

whose k - 1 derivatives interpolate to zero at interior mesh points of A . Finally, ~(x)minimizes the integral

s”,

(5.6.1)

f (nl(x)}zdx.

sd(x)

Consider now any two splines Sd(x) and on A such that on A and is of type 1’, type 11’, or periodic. Since Sd(x) - S,(x) is a spline interpolating to x(x) on A , we must have

sA(x)- S,(x) vanishes J

b a

{Sy)(x) - ST’(X)}~dx = 0.

(5.6.2)

sA(x)

From the continuity of the integrand, we infer that S,(X) and differ by a polynomial of degree n - 1 which vanishes on A and is of type 1’, type 11’, or is periodic! We have the following theorem.

< xN = b and f(x) be Theorem 5.6.1. Let A : a = xo < x1 < given. Then in each type I equivalence class there is at most one spline of interpolation to f ( x ) on A. I f f ( a ) = f ( b ) , then there is at most one periodic spline of interpolation to f ( x ) on A. Finally, if N > n - 2, then in each type 11equivalence class there is at most one spline of interpolation tof ( x ) on A . Also, if S,(x) and sd(x) are two polynomial splines of degree 2n - 1 on A both of deficiency k(k < n) such that Sd(xi)= sA(xi)(i= 0, 1,..., N ) , if Sy)(xi)= sy)(xi)(i= 1, 2 ,..., N - 1; a = 1, 2,..., k - 1) and S,(X) - sd(x) is of type 1‘, type 11’, or periodic, then we can argue just as before that (5.6.2) holds. This establishes Theorem 5.6.2, of which Theorem 5.6.1 is a special case.

Theorem 5.6.2. Let A : a = xo < x1 < -.. < xN = b and f ( x ) be given, where f ( x ) has k - 1 derivatives at each interior mesh point of A . Then in each type I equivalence class there is at most one polynomial spline s A ( f ; x) of degree 2n - 1 and deficiency k(k n) which interpolates to f ( x ) on A and such that S i p ) ( f ;x) = f ( a ) ( x )( a = 1, 2, ..., k - 1) at the interior mesh points of A . Similarly, if f ( a ) = f ( b ) , then there is at most one periodic spline of deficiency k satisfying these interpolation requirements. Finally, in each type 11 equivalence class there is at most onesuch spline provided k(N - 1) >, n - 2.

<

5.7. Defining Equations T h e representation obtained for polynomial splines of odd degree in Chapter I V is complex, as Eqs. (41.5)-(4.1.13) reveal. We now proceed to replace these equations by a system of equations analogous to those obtained in Section 3.7.

5.7.

161

DEFINING EQUATIONS

Let u j ( x )= ( l / ( j - l)!)xj-I ( j = 1 , 2 ,..., 272). Then u:P)(O)= and the functions ui(x) constitute a fundamental system of solutions of the differential equation Dznf

(5.7.1)

= 0.

Thus, any solution of (5.7.1) is of the form (5.7.2)

Since, for a given mesh A , a polynomial spline S,(x) satisfies (5.7.1) on each open mesh interval xiPl x xi (i = 1 , 2, ..., N), we have

< <

S,(x)

c 2n

=

CijUj(X

- xi-1).

j=1

This representation is unique, since the functions u j ( x ) are linearly independent. Consequently, such a set of cii (i = 1 , 2,..., N;j = 1 , 2,..., 2n) uniquely defines a spline SA(x),and, conversely, S,(x) uniquely defines a set of cii if S,(x) exists. Utilizing the representation (5.7.2) together with both the continuity and interpolation requirements imposed on type I, type 11, and periodic splines, respectively, we are led to the following system of equations for determining the cij and consequently S,( Y ; x).

A,

,

=

(5.7.3.1)

,=I.II)

0 0

...

A p =

0 B,

(5.7.3.2)

v. POLYNOMIAL

162

SPLINES OF ODD DEGREE

, (5.7.3.3) h.=z I

1 -1

0 0

0 -1

B,

=

...

0 0

..'

0

..'

... 0 0

0 0

0

0

...

... - 1

...

...

0

... 0 ... 0

"'

...

v

0

...

0 0 1 0

\

... ... 0 ... .o ...

I

c1.n

=

0

0

0

...

(5.7.3.4)

... 0 0.

1

...

...o ...o

...

\

I-1

... 0 ... 0

... ...

... 0 ... 0

--2.

0-

0

0 0 -1

2n '1

0

t

v n

1 0 0 1

o... o...

(5.7.3.5)

...

...o ...o

1 -r

2n

,

... 0 ... 0

,

(5.7.3.7)

,

.

Yo (1) Yo

Y1

Yo

(n)

(n-1)

163

0

Yo

hn-2)

Yo

Yo

Y1

Y1

0

0

0

Y2

0 0

0

Yz

YZ

0

0 Va 0

C =

I

Yr

=

YII =

I

0

0

YN-1

YN-i

0

YP =

0

0

0

" 0 0

0

YN

YN

(n)

(1)

YN

YN

(n-1)

,

YN

5.7.3.9)

.

(5.7.3.10)

(zn-2)

0

(5.7.3.11)

(5.7.3.12)

YN

When the spline S,( Y; x) is of deficiency k, the preceding system of equations should be modified as follows:

(5.7.4.1)

164

k -

B,

/

1

0 0

0

1

...

-.

0

...

=

0

0

Ol

0 0

O ... O

1

0 0 0 ,

I(5.7.4.2)

-1

0 0 0 -1 0 - 0 0 0

---

--

00-1

0

O O O J

2n - k r

-

Yo (1) Yo

(n-1)

Yo

Y 1 (k-1) Y 1

0

0 Y Z (k-1) Y Z

0

YI

=

0 YN-1 (L-1) YN-1

0

0 YN

(1)

YN

- (n-1)

YN

(5.7.4.3)

(5.7.4.4)

(5.7.4.5)

(5.7.4.6)

5.8.

165

EXISTENCE

I n the next section, we investigate the invertibility of the matrix A, upon which the existence of S A ( Y ; x) depends. The equations for the periodic spline are for the case: S2’(f ;a ) = .fU’(a)( a = 0, 1,..., k - l), S y ’ ( f ;a ) = Sy’(f ; b ) ( a = 0, 1,..., 2n - k - 1).

5.8. Existence In Section 5.6 we established the uniqueness of polynomial splines of odd degree, and in Section 5.7 we established that to every polynomial spline of odd degree S,(X)there corresponds a unique set of coefficients cij (i = 1, 2,..., N ; j = 0, 1,... 2n - 1). Consequently, if the equation A&

(5.8.1)

=0

has more than the null solution, there are two distinct null splines; this would contradict Theorem 5.6.1 or Theorem 5.6.2. Thus, A;l exists, and we have Theorems 5.8.1 and 5.8.2, which supplement Theorems 5.6.1 and 5.6.2, respectively.

Theorem 5.8.1. Let A : a = xo < x1 < < xN = b and f ( x ) be given. Then in each type I equivalence class there is a unique spline of interpolation to f ( x ) on A . If f ( a ) = f ( b ) , then there is a unique periodic spline of interpolation to f ( x ) on A . Finally, if N > n - 2, then in each type 11 equivalence class there is a unique spline of interpolation to f ( x ) on A. Theorem 5.8.2. Let A : a = xo < x1 < * - * < xN = b and f ( x ) be given, where f ( x ) has k - 1 derivatives at each interior mesh point of A. Then in each type I equivalence class there is a unique polynomial spline SA(f; x) of degree 2n - 1 and deficiency k (k n) which interpolates to f ( x ) on A and such that S y ) ( f ;x) = f ( a ) ( x )( a = 1, 2, ..., k - 1) at the interior mesh points of A. Similarly, if f ( a ) = f ( b ) , then there is a unique periodic spline of deficiency k satisfying these interpolation requirements. Finally, in each type 11 equivalence class there is a unique spline provided n - 2. k(N - 1)

<

The uniqueness theorems and the existence theorems obtained in this chapter follow directly from the minimum norm property and could have been established prior to the proof of the best approximation property. This is important, since the proof given in Section 5.5 of the latier property requires the existence of the auxiliary spline SA(f - SA ; x). An alternative proof of the best approximation property could be given, however, which would not use the auxiliary spline, but there would be a definite loss in simplicity of presentation.

166

V.

POLYNOMIAL SPLINES OF O D D DEGREE

I n the case of the cubic spline, the existence theorem of Chapter I11 is not so strong as that obtained in Chapter I1 by other methods; in the case of polynomial splines of odd degree, however, these methods, as pursued in Chapter IV, lead to an existence theorem requiring a nearly uniform mesh. Consequently, Theorems 5.8.1 and 5.8.2 are of major importance.

5.9. Convergence of Lower-Order Derivatives Letf(x) be a function in %%(a, b), and let S,(f; x) be a spline of interpolation to f(x) on a mesh A : a = xo < x1 < * * * < xN = b. Repeated application of Rolle's theorem tells us that at least once in every (Y consecutive intervals S ; ) ( f ; x ) interpolates to f(a)(x) (0 01 < n). If 2- is such a point of interpolation,

<

f ( - ) ( x ) - S F ) ( f ;x)

{ f ( a + l ) ( x) Sgl"+I)(f;x)} dx.

=

(5.9.1)

501

<

Here the interval of integration can be chosen so that I $a - x 1 +(a 1) 11 A /I whenever such interpolation points lie both to the left and to the right of x, since can be selected on either side of x. If this is not the case, the factor 8 generally must be omitted. When considering a sequence {A,} of meshes with Ij A , /I + 0 as N -+ co and when x is not either a or b, the factor of can be included as soon as N is sufficiently large so that x lies between such interpolation points. When, in particular, f ( x ) - S,(f; x) is of type I' or both f ( x ) and S,(f; x) are in X p n ( a ,b), the factor need never be omitted. Let 01 = n - 1. Then we have

+

+

+

(5.9.2.1)

where JZ

= Jb a

and B is either result that

8 or

{f'"'(X)

-

Sl;"'(f;x)}2 dx

(5.9.2.2)

1. Repeating this argument, we obtain the general

l f ( ' y x ) - S$'(f; x) 1

<J

*

n1/2(n -

' B(Zn-2m-1)/2

1)

*.*

(a

+ 1)

. II A /I( 2 n - Z a - l > / 2

(5.9.3)

5.9.

167

CONVERGENCE OF LOWER-ORDER DERIVATIVES

<

for 0 a < n. Theorem 5.9.1, which follows, is now immediate if one observes that Minkowski's inequality, together with the minimum norm property (Theorem 5.4.1), implies that

Theorem 5.9.1.

Let f ( x ) be in X n ( a , b), and let { A N : a = b} be a sequence of meshes with 11 d, Ij -+ 0 as N -+ 03. Let S,,(fi x) be a spline of interpolation to f ( x ) on AN ( N = 0, 1,...) satisfying one of the conditions (a) f(x) - S d N ( f ;x) is of type I' ( N = 0, 1,...), (b) f(x) and Sd,(f; x) are periodic ( N = 0, 1,...), (c) S,(f; x) is of type 11' ( N = 0, 1, ...). Then we have xoN

<

< ,x;

f'"'(X)

Iqu(x) 1

=

=

+

SY$((f; x)

< (n>""n

-

1)

( a = 0, 1,..., n - l),

7&)

* * * (Lx

+ 1)

*

2-'2n-7=1)'2

*

(5.9.5.1)

11 A N 11(2n-Zm-1)/2 (5.9.5.2)

When condition (c) is assumed, the factor 2--(2n--2u-1)/2 must be omitted for x = a and x = b. Finally, i f the condition (d) A,C A N and S,,(f; x) - Sdo(f; x) is of type 11' ( N = 1,2, ...), is satisfied, then (5.9.5.1) and (5.9.5.2) are valid i f

is replaced by

We could easily formulate an analog of Theorem 5.9.1 for the case of type I or type I1 splines of deficiency k. I t is perhaps somewhat more illuminating, however, to consider polynomial splines of deficiency k such that f ( = ) ( x i )= S y ) ( f ;xi)( a = 0, 1,..., k - 1) not only at the interior mesh points of d but at a and b as well. I n order to define the spline completely in the nonperiodic case, we require in addition S F ) ( f ;xi) = 0

( a = n, n

+ 1 ,..., 2n - k

-

1; i = 0, N ) . (5.9.6)

In the periodic case we require S y ) ( f ;a ) = S y ' ( f ;b) for CL = 0, I , ..., - k - 1. We refer to these splines, both nonperiodic and periodic, as splines of interpolation of type k.

2n

168

v. POLYNOMIAL

SPLINES OF ODD DEGREE

For splines of interpolation of type K the first integral relation is valid, and, consequently, we have the minimum norm property, uniqueness, existence, and the best approximation property. These four properties allow us to obtain the counterpart of Theorem 5.9.1 by a parallel argument, which we omit.

Theorem 5.9.2. Let f ( x ) be in X n ( a , b), and let { A , : a = < ..* < xmN = b) be a sequence of meshes with II A N 11 + 0 as N -+ a. Let SdN(f;x) be a spline of interpolation to f ( x ) on A N satisfying one of the conditions, (a) SdN(f;x) is of type K ( N = 0, 1, ...), (b) f ( x ) is in -Xpn(a,b), and SAN(f; x) is a periodic splzhe of type K ( N = 0, I,...). Then XoN

f ' " ' ( ~= ) SF;( f;X)

+ Y,(X)

(a = 0,1, ..., n

-

I),

.

(5.9.7.1)

where

(5.9.7.2)

When condition (a) is assumed, the factor 2--(2n-201-1)/2 must be omitted for x = a and x = b if a > K - 1. If n - 2k 2 < 1 in (5.9.7.2), it is replaced by 1 .

+

+

1) in (5.9.5.2) is T h e reason that the factor (n)'lz(n- 1) ( a 2)1/2(n- 2K l ) ! / ( a - 2K 2)! in replaced by the factor ( n - 2K (5.9.7.2) is that Sizk-l)(f; x) interpolates to f (2k-1)(x)at least once in x) every mesh interval when S,( f ; x) is a spline of type K . Thus, SiZk-l)(f; in this situation possesses interpolation properties normally associated with the first derivative. I n interpreting (5.9.7.2), we interpret m! as 1 when m is a nonpositive integer.

+

+

+

5.10. The Second Integral Relation T h e relations obtained in Sections 3.9 and 3.10 can be extended to polynomial splines of degree 2n - 1 in a straightforward manner. Let (5.10.1) and

llf I/ = (f7f)1/z*

(5.10.2)

5.10.

THE SECOND INTEGRAL RELATION

I n addition, let A : a = xo < x1 < be a polynomial spline of degree 2n may write

Sb

{ f ( n ) ( x) sF)(x)}2dx

=

< x N = b be given, and let S,(x) -

1 on A . If f ( x ) is in Z Z n ( ab), , we

CN Jx'

i=l

a

169

{ f ( " ) ( ~ )- S ~ ) ( X dx. ) } (5.10.3) ~

Xi-1

Next we integrate each of the integrals in the right-hand member of (5.10.3) by parts n times and obtain

f

1( f b

a

-

- SA) D2nf dx.

(5.10.5)

T h e identity (5.10.5) is the analog of the identity (3.9.4). Under a variety of conditions on f ( x ) - S,(x), the identity (5.10.5) reduces to the relation

which we call the second integral relation. We observe that, for a spline of interpolation S d ( f ; x ) to f ( x ) on d of type k, the second integral relation normally is not true. We have, however, the following two theorems.

Theorem 5.10.1. Let A : a = xo < x1 < < x N = b and f ( x ) in X Z n ( ab) , be given. If S,( f ; x ) is a polynomial spline of degree 2n - 1 which interpolates to f ( x ) on A and S d ( f ;x ) satisfies one of the conditions, (a> f ( 4- S4.f;x> is of type I',(b) f ( x > - S,(f; x ) is of tYPeII', ( 4 f ( x ) and S d ( f ;x ) are periodic, then

Ilf

- S A ,Ila~ =

J-b (f a

- SA.,) . D2nfdx.

v.

170

POLYNOMIAL SPLINES OF ODD DEGREE

Theorem 5.10.2. Let A : a = xo < x1 < < x N = b and f ( x ) in X Z n ( a ,b) be given. If S,(f; x) is a polynomial spline of degree 2n - 1 and deJiciency k which interpolates to f ( x ) on A , satisjies the interpolation conditions . - a

S'm'(f;xi)

( a = 1, 2 ,..., k

=f'*'(~i)

-

1; i

=

1, 2 ,..., N - I), (5.10.7)

and one of the conditions, (a) f ( x ) - S,( f ; x) is of type 1', (b)f ( x ) - S,( f ; x) is of type 11', (c) f ( x ) and S,(f; x) are periodic, then

llf

-

S A ,/I2~ =

Jb

a

(f - S A , ~ Dznf ) dx. *

As we have already pointed out, the second integral relation generally is not valid for splines of type k. If we consider splines satisfying the slightly modified end conditions S y ( f; Xi) = f ' " ' ( X i ) (OL

= 0 , 1 , ..., k - 1;

OL

= n , n + 1,..., 2 n - k -

1; i = O , N ) ,

(5.10.8)

the second integral relation is again valid. I n this case, however, the minimum norm property fails, but the best approximation property holds. We refer to splines of interpolation of this form as splines of modijied type k . A type I1 spline is a spline of modified type I.

5.1 1. Raising the Order of Convergence I n this section, we proceed very much as in Section 3.10. From (5.9.3), it follows that If(,)(x) - Sp)(f;X)

1

< K,

1s

b

*

a

{f(")(x) - S y ) ( f ;x)}z dxt

1 I2

I/ d /l(zn-z,-1)/2

(5.11 .I) for a suitable choice of the constant K, . If the second integral relation is valid, (5.11.1) is equivalent to

I f'm)(4 - Sp)(f; 4 I

hence, we have

5.11.

171

RAISING THE ORDER OF CONVERGENCE

As in Section 3.10, we set a: = 0 and solve (5.11.2) for supz I f ( x ) - S4(f;x)I, which is possible except in the trivial case when f ( x ) = S,(f; x). Thus, S;P

).(fi

- Sd(f;

X)

1

< KO2

*

Vab[f'2n-1'] * 11 A

1 2n-1,

(5.11.3)

which, together with (5.11.2), implies I f ( , ) ( x ) - SSp)(f;X)

1

< K,

KO* Vab[f(2n-1)] * 11 A

(5.11.4)

l/2n--ol-1.

We are now able to reformulate Theorems 5.9.1 and 5.9.2.

Theorem 5.1 1.1.

Let f ( x ) be in A?2n(a, b), and let { A , : a = x o ~ < " ' < x ~ =b} h'

be a sequence of meshes with 11 A,I( -+ 0 as N + CO. Let {S+(f; x)} be a sequence of splines of interpolation to f(x) satisfying one of the conditions (a)f(x) - SAN(f; x) is of type I' ( N = 0, I,...), (b) f(x) and S+(f; x) are periodic ( N = 0, 1,...), (c) f ( x ) - S,(f; x) is of type II' ( N = 0, 1 ,...). Then (5.11.5.1) (a = 0, 1,..., n - I), f ' " ' ( ~ )= SSp'(f;X) T,(x) where

+

1 T,(X) 1

< n(n - 1)

(a

+ I)[@ - I ) ! ]

. Vab[Cf(2n-l)] , (1 A ,

*2-(2n-a-1)

/12n--ol-la

(5.1 1.5.2)

W h e n condition (c) is satisfied, the factor 2-(2n-a-1)must be omitted for x = a and x = b.

In reformulating Theorem 5.9.2, we employ splines of modified type

k rather than splines of type k.

Theorem 5.11.2, L e t f ( x ) be in X Z n ( a ,b), and let {A,

: a = xoN

<

< X:

N

= b}

be a sequence of meshes with /I A , 11 -+0 as N + 00. Let { S d N ( fx)} ; be a sequence of splines of interpolation to f ( x ) satisfying one of the conditions (a) SdN(f;x) is of modijied type k ( N = 0, 1,...), (b)f(x) is in X p ( a , b), and S d N ( fx) ; is a periodic spline of type k ( N = 0, 1,...). Then f ' " ' ( ~ )= SFA(f;X)

+ T~(X)

( a = 0,I ,

...,n - l),

(5.11.6.1)

172

v. POLYNOMIAL

SPLINES OF ODD DEGREE

where

must be omitted when W h e n condition (a) is satisfied, the factor 2--(2n--a-1) = a or x = b and 01 >, k. If n - 2k 2 < 1 in (.5.11.6.2), it is replaced by 1.

+

x

5.12. Convergence of Higher-Order Derivatives Let A : u = xo < < xN = b be given, and let f ( x ) be in Z Z n ( u ,b). I n addition, let S,,(f; x ) be a polynomial spline of degree 2n - 1 interpolating to f ( x ) on d such that

I f ( x ) - S,(f;x) 1

< Ko2

*

Yab[f'2"--1']

*

11 d p

-

1

(5.12.1)

is valid. For the equally spaced difference quotients 6;A,/[xi-1 , Xi]

where 6$A.,[xi-1, xi] xu,i with xi--l xu,$

<

=

(i = 1, 2,...,N; a Sd(f;xi-l)

= 0,

=f(xi-J,

< xi such that

,xi]= S $ ) ( f ;xUmi) (i

=

1, 2 ,..., N; a

1 ,..., 2n - l),

we can find quantities = 0,

1 ,..., 2n - 1). (5.12.2)

As a consequence, the equations

can be used to determine the coefficients cii for small values of since

11 d 11,

zn

as 11 d )I + 0 uniformly with respect to both the number and location of the mesh intervals [ x i - l , x i ] . Moreover, since 6;[a, b] is a linear operation with respect to f,

I s".d,,[xi-l xi1 - 6$Xi-, 7

9

xi1 I = I 6>A,,-f[xi--l xi1 I . 9

(5.12.4)

5.12.

173

CONVERGENCE OF HIGHER-ORDER DERIVATIVES

I t follows that I S > A , t [ ~ i - l , xi] - S;[X~-~ , X] 1

< (2aRA)a

(i = I , 2,..., N ;

3

K,2

*

V/[f'2n-1']

a = 0 , 1,..., 2n

*

-

/I I12n- ar-1 I),

(5.12.5.1)

where (5.12.5.2)

Theorem 5.12.1.

Let f ( x ) be in 3Y2n(a,b) and { A , :a

= xON

< - I . < x,"

N

=

b}

be a sequence of meshes such that 11 A , 11 + 0 as N + co.Let R A N , defined by (5.12.5.2/, be bounded with respect to N , and let (SAN( f;x)) be a sequence of polynomial splines of degree 2n - 1 which interpolate to f(x) on corref ;x) sponding meshes A , and which satisfy one of the conditions (a) f ( x ) - SAN( is of typeI' ( N = 1, 2,...), (b) f ( x ) - SdN(f; x ) is of typeII' ( N = 1, 2,...), (c) f ( x ) and s A N ( f ; x ) are periodic ( N = 1, 2, ...). Then f ( = ) ( x )= S(=)f;x) AN

+ O(1l d,

)12n- u-1)

( a = 0, I , ..., 2n

-

1) (5.12.6)

uniformly for x in [a, b]. The proof is essentially the same as that of Theorem 3.11.1. By a completely parallel argument, we have Theorem 5.12.2, which should be compared with Theorem 5.11.2. Where Theorem 5.11.2 applies it is sharper, particularly when k > 1. Details of the proof of the final assertion of Theorem 5.12.2 can be found in Section 6.12.

Theorem 5.12.2.

Let f ( x ) be in 3Y2n(a,b) and (A, :a

= xoN

< ... < x," N

=

bf

be a sequence of meshes such that 11 A , 11 + 0 as N 3 co.Let R A N , defined by (H2.5.2), be bounded with respect to N , and let {SAN( f ;x)>be a sequence of polynomial splines of degree 2n - 1 which interpolate to f ( x ) on corresponding meshes A , and which satisfy one of the conditions (a) SAN(f;X ) is of modijied type k ( N = 1, 2, ...), (b) f ( x ) is in &';"(a, b) and sA,(f; x) is a periodic spline of type k ( N = 1,2,...). Then we have, uniformly f o r x in [a, bl, ( a = 0, I, ..., 2n - K). (5.12.7) f ( a ) ( x )= SF$(f;x) + O(ll A , l12n--or--k) Moreover, limN+coSyA(f;x)

=f

(a)(,)

uniformly in x for

01

= 0,

1,..., 2n - 2.

v. POLYNOMIAL

174

SPLINES OF O D D DEGREE

5.13. Limits on the Order of Convergence T h e discussion of limitations on the convergence of cubic splines contained in Section 3.12 carries over essentially unchanged to polynomial splines of odd degree. I n general, we have f ( a ) ( ~ )=

Spi(j;X)

+ O(1l

A N

1 212-a-1)

(a = 0,

1,...,2n - 1). (5.13.1)

For a = 0, I , ..., n - 1, no restrictions are imposed on the meshes, but I , ..., 2n - I , the mesh parameters RANmust be bounded for a = n, n as a function of N . For uniform spacing we obtained in the periodic case (cf. Section 4.6, Theorem 4.6.3) the stronger result that

+

pyx)

=

S'e' A , ( f ; x)

+ O(1i d,

( a = 0, 1,..., 2n - 1).

l / 2 n- a )

(5.13.2)

I n the case of (5.13.1), the rate of convergence is proportional to Vab[f(2n-1)], whereas for (5.13.2) the rate of convergence is proportional to IIf(2n)/ I m for f ( x ) in CZn(u,b). T h e rate of convergence in (5.13.2), insofar as it depends on 11 A , 11, cannot be improved. We have, in fact, the following theorem, the proof of which is the same as that of Theorem 3.12.1.

Theorem 5.13.1. Let {A,} be a sequence of meshes with 11 A N 11 -+ 0 as N 4 co,and let RANbe bounded with respect to N . Let f ( x ) be in C2n(u,b) and p > 0. If, uniformly f o ~x in [u, b],

f(.)

SLl,(f;

then

).

Dznf

+ O(l1

1/ 2 n+p) ,

(5.13.3)

= 0.

REMARK5.13.1. T h e splines involved in Theorem 5.13.1 need not be of deficiency one; the key hypothesis is (5.13.3). 5.14. Hilbert Space Interpretation We continue here the discussion begun in Section 3.13. T h e class

X n ( u , b), under the pseudo-inner product ( f ,g)

=

jbf(W . g ' n ' ( 4 dx a

(5.14.1)

is a Hilbert space if we identify functions that differ by a polynomial of g), degree n - I ; without these identifications and the inner product (f,

5.14.

HILBERT SPACE INTERPRETATION

175

A : a = x,, < x, < < xN = b be given, and let F,(n, k) denote the family of polynomial splines on A of n). As a linear subspace of degree 2n - 1 and deficiency k (k X n ( a , b), FA(n,k) has dimension k(N - 1) 2n, and as a Hilbert space, where splines differing by a polynomial of degree n - 1 are identified, it has dimension k ( N - 1) + n. If PA(n,k) denotes the family of periodic polynomial splines on A of degree 2n - 1 and deficiency k, then PA(n,k) is a subspace of FA(n,k ) . As a linear space (without any identifications), P,(n, k) has dimension N * k ; as a Hilbert space (with splines differing by a constant identified), PA(n,k) has dimension Nk - 1. If d, refines A , , FAl(n,k) is a subspace of Fdz(n,k), and PAl(n,k) is a subspace of P,,(n, k). Since they are finite dimensional, FA(n,k) and PA(n,k) are always closed subspaces. If A , C A , , then [FA,(=,k) - FA,(n,k)] and [PA,(n,k) - P J n , k)] denote the splines in FA2(n,k) or PAz(n,k), respectively, whose defining values (including any derivatives*) o n d , are zero. We employ similar notation for other spaces when required. T h e orthogonality of the component spaces in the decompositions in the remainder of this section is demonstrated in Section 5.15. Observe that the linear spaces [FAl(n,k) -FAi-l(n, k)],etc., unlike the spacesF,(n,k), are unaffected when systems differing by a polynomial of degree n - 1 are identified. Thus, we can regard elements in these spaces as functions rather than equivalence classes even after identifications are made. Consider now a sequence of meshes { A N } on [a, b] with A N C ( N = 1, 2, ...). I t is true that X n ( a , b ) is simply a linear space. Let

<

FAN(%

+

k) = FA,(% k) 0[FAz(% k) - FAl(% @ .'* 0 k, - FdN - l( n >

41 '11,

(5.14.2.1)

m

and

* Here and in the remainder of this chapter we take these defining values to be the entries in (5.7.3.10), (5.7.3.11), or (5.7.3.12) associated with A , .

v. POLYNOMIAL

176

SPLINES OF ODD DEGREE

respectively. Clearly, we have the inclusion relations PA,(% k) c FA,(% k),

(5.14.4.1)

c X " ( a , b).

(5.14.4.2)

FA&, K )

I n Section 5.15, we establish for k # 0 that F,,(n, k) = %"(a, b ) and that P,,(n, k) = %"(a, b ) provided 11 A , 11 -+ 0 as N -+ co. Since, by definition, F,,(n, k) and P,,(n, k) are closed with respect to the norm

llfll = (fLW2,

(5.14.5)

this is equivalent to showing that F,,(n, k) is dense in %%(a, b ) and P,,(n,k) is dense in %%(a, b). First, however, we single out two additional subspaces of F,(n,k) which are of considerable importance. T h e first of these subspaces is the family of type 11' polynomial splines on A of degree 2n - 1 and deficiency k; the second is the family of type k polynomial splines on A of degree 2n - 1." We denote these subspaces by FL(n, k) and T,(n,k), respectively. Observe that Fi(n, K ) C T,(n,k) CF,(n, k). Given a sequence of meshes {A,} on [a, b] with A , C A,+l ( N = 1, 2, ...), the subspaces Pi(=, k) and T,(n,K) allow us to define the infinite direct sums Fi,(n, k) and T,,(n,k). We define Fi,(n, K ) as F>,(n,

4

c 0[Fi,(n, k) -FiN-Jn, 4 1 m

=F

p , k) 0

(5.14.6.1)

N--2

and T,,(n,k) as m

T,,(n> ')

=

Td,(n,

')

0

N-2

0

')

-

')I*

(5*14.6*2)

We establish in Section 5.15 that, for k # 0, T,,(n, k) = F i m (n,k) = %"(a, b),

if 11 A N 11

+0

(5.14.7)

as N + 00. 5.15. Convergence in Norm

Theorem 5.15.1. Let { A , : a = x o N < ... < , :x = b} be a sequence of meshes with A , C A,,, ( N = 1, 2,...), 11 d, 11 + 0 as N + co, let * A spline is of type k if for some function f ( x ) it is a spline of interpolation to f ( x ) of type k.

5.15.

CONVERGENCE IN NORM

177

f ( x ) be in %lz(a, b), and let {SA,(f;3)) be a sequence of polynomial splines of degree 2n - 1 and deficiency k (k a ) with S$;(f; xiN) = f(u)(xi”) (i = 1, 2,..., mN - 1; a = 0, 1,..., k - 1) f o r each N . If, in addition, one of the conditions, (a) f ( x ) - SdN(f; x) is of type I’ ( N = 1, 2, ...), (b) S , , ( f ; x ) i s o f t y p e I I ’ ( N = 1 , 2,...) , ( c ) S d N ( f ; x ) i s o f t y p e k ( N1=, 2,...), (d) f ( x ) and SdN(f; x) are in ,Xplz(a,b), ( N = 1,2,...), is satisfied, then Ilf - SdN.,ll + 0 as N + 00.

<

Proof. Since the minimum norm property and the first integral relation are valid, the proof is a replica of the proof of Theorem 3.14.1. Here the nth derivative plays the role of the second derivative in the proof of Theorem 3.14.1 and the ( n - 1)th derivative the role of the first derivative.

REMARK5.15.1. T h e norm of functions in Theorem 5.15.1 is the Hilbert space norm; i.e.,

llfll

=

(f?.w2.

(5.15.1)

I n any Hilbert space, the concept of an infinite direct sum is meaningful. As indicated earlier, if Vi is orthogonal to Vj (i # j ) and (5.15.2)

then V is the smallest closed subspace containing all the component spaces V,. Moreover, we can choose a basis for V which is simultaneously the extension to V of a basis for each V,. Consequently, Theorem 5.15.2 follows from Theorem 5.15.1, since a dense subspace that is closed must be the whole space.*

Theorem 5.15.2, Let {A,} be a sequence of meshes on [a, b] with ( N = 1, 2, ...) and 11 A , // + 0 as N -+ co. Then

d, C A,,,

FAm(n,K)

= Fim(n,K) =

T,m (a, K)

= Y”(u, b),

P,,(n, k) = %“(a, b).

(5.15.3.1) (5.15.3.2)

I n Section 3.14, we pursued essentially the same approach, and we established for the analogs of our direct sums FAm(n,K ) , Flm(n,K), TAm(n,K ) , and P,,(n, k) that the component spaces are mutually orthogonal so that the indicated decompositions are decompositions in the Hilbert space sense. This is still true; we have, in fact, the following lemma, which is an extension of Lemma 3.14.1.

* A proof that PAm(n,K) is dense in %“(a, 6 ) is given in Section 6.14.

178

v.

POLYNOMIAL SPLINES OF ODD DEGREE

Lemma 5.15.1, Let A , and A , be two meshes on [a, b] with A , C A , , and let Sdl(x) and SA2(x)be two polynomial splines on A , and A , , respectively, each of degree 2n - 1 and deficiency k (k n). If Sk)(x) ( a = 0, 1,..., k - 1) vanishes a t the interior mesh points of A , and, in addition, one of the conditions, (a) SAl(x)is of type II’, (b) SA2(x)is of type If,(c) SAl(x)and SA2(x)are of type k with S$)(a) = S k ) ( b ) = 0 ( a = 0, 1,..., k - l), (d) SAl(x)and SA2(x) are in X p n ( a , b) with S x ) ( a )= 0 ( a = 0, 1,..., k - l), is satisfied, then (SAl , SA,) = 0.

<

Proof.

Let A , be defined by a

xo

=

< x, <

< xN = b.

Then,

If we integrate by parts n times, we obtain

and the lemma follows. Consider now any one of the infinite direct sums FA,(n, k), Fim(n,k), TAm(n, k), or PA,(n, k); for instance, FAm (n,

4

m

= FAl(%

k) 0

10

N=2

[FAN(%

4 -FA,-,(% 41.

(5.15.4)

I f S A N ( x )is in [FAN(n,k) - FAN-l(n,k)], then Syi(x) (ci = 0, 1,..., k - 1) vanishes at the interior mesh points of ANPI and is of type 1’. T h u s Lemma 5.15.1 applies, and we can conclude that (5.15.4) is an orthogonal decomposition.

Theorem 5.15.3, Let {A,} be a sequence of meshes on [a, b] with 4, C AN+l ( N = 1, 2, ...) and such that 11 A , 1) -+ 0 as N -+ CO. Then the indicated injinite direct sums FA,(% k )

= FAl(n,k)

0

2 0 FA,(^, k)

N=2

-

FAN-l(n, k)],

(5.15.5.1)

5.16.

179

CANONICAL MESH BASES AND THEIR PROPERTIES

are orthogonal decompositions with respect to the inner product (5.14.1), and all are identical with X n ( a , b).

5.16. Canonical Mesh Bases and Their Properties Canonical mesh bases were introduced in Section 3.15 for X 2 ( a ,b) and X p 2 ( a ,b). Similar bases can be obtained for X n ( a , b) and Xpn(a, b ) ; in fact, we now obtain such bases for FAm(n,k), Fi,(n, k), TA,(n,k), and PAm(n, k). Consequently, in view of Theorem 5.15, we also obtain a variety of orthonormal bases for both X n ( a , b) and S P n ( a ,b). Since the constructions are essentially the same, we consider, explicitly, only the construction of a mesh basis for F,,(n, k). Even here, we simplify the procedure over that of Section 3.15 by limiting ourselves to a single straightforward enumeration of mesh points and thereby omit mesh bases that are not canonical. Let {A,) be a sequence of meshes on [a, b] such that A , C A,+, ( N = 1, 2, ...). Let M be the set of all distinct mesh points contained in the meshes A , , excluding the mesh points of A , . Give M the enumeration { M :P, , P, ,...}, where mesh points are enumerated starting with the mesh points of A , and counting from left to right, then proceeding to the mesh points of A , , etc., in each case passing over mesh points previously enumerated. Now define a new sequence of meshes {n,} ( m = 0, 1,...) where T,, = A , , 77, = A , u {P,}, and, in general, n, = n,-, u {P,}. We assume that we are given an orthonormal basis for Fdl(n, k ) , and we construct an orthonormal basis for [FA,(n, k ) - FAl(n,A ) ] , thus extending the original basis to an orthonormal base for F,,(n, k). For each m ( m = 1, 2,...), we let h,,(x) ( j = 1, 2,..., k) be type I' polynomial splines on T, of degree 2n - 1 and deficiency k such that each hgi (x) ( a = 0, 1, ..., k - 1) vanishes on T,-, and the hmj (x) are orthonormal. We can take g,.,(x) ( j = 1, 2, ..., k) to be the typeI' splines on d, such that gC)(x) (a = 0, 1,..., k - 1) vanishes on A,-, and ggj(P,) = a;+'; the Gram-Schmidt process now yields the desired hmj(x). If for i = k(m - 1) + j , where k ( m - 1) < i km, we set

<

Si(n,K ; X) = hmj(x)

(WZ =

1, 2,...; j

=

1, 2,...,k ) ,

(5.16.1)

we obtain the desired orthonormal basis for [FA,(n,k ) - FAl(n,A ) ] . Indeed, the subset {Si(n,k; x) I i = 1, 2, ..., km} is an orthonormal basis for [FWm(n, k ) - FT0(n,k ) ] ( m = 1 , 2,...), and for each N ( N = 1 , 2,...) the Si(n, k ; x) which, together with their first k - 1 derivatives, do not vanish at every mesh point of A , , constitute an orthonormal basis for

v.

180

POLYNOMIAL SPLINES OF O D D DEGREE

[FA,(n, k) - FAl(n, k)]. When there is no ambiguity, we denote Si(n, k ; x) by S,(x) and FA(n,k) by FA . We call the set of Si(x) (i = 1, 2,...), together with the preassigned orthonormal basis for FA, , a canonical mesh basis for FA, ; their analogs for F i , , T A , , and PA, are called canonical mesh bases for these spaces. We have the following theorem, which is now immediate from Theorem 5.15.

Theorem 5.16.1. Let {A,} be a sequence of meshes on [a, b] with A , C A,,, ( N = 1, 2, ...) and 11 A , 11 -+ 0 as N -+ 00. Then FAm(n,k), Fi,(n, k), and TA,(n, k) have canonical mesh bases that are orthonormal bases f o r X n ( a , b ) ; PA (n, k) has a canonical mesh basis that is an orthonormal basis f o r Xpn(< b) and X n ( a , b).

REMARK5.16.1. I n the preceding theorem, both X n ( a , b) and X p n ( a , 6 ) are meant to be interpreted as Hilbert spaces under the inner product (f,g) defined by (5.14.1). However, to be precise, ,Xpn(a, b) is a pre-Hilbert space. We formulate next the analog of Lemma 3.15.1 which furnishes extremely useful information regarding the magnitude of Sp)(x) ( a = 0, 1,..., n - 1).

Lemma 5.16.1. Let {Si(n,k ; x)} be a canonical mesh basis f o r k)], [Fi,(n, k) - FLl(n, k)], [TA,(n, k, - T A l ( n , k)]> k) or [PA,(n, k) - PAl(n,k)] determined by a sequence of meshes {A,} with A, C A,+, ( N = 1, 2,...). Let {T,} ( m = 0, 1,...) be the related sequence of meshes used in the construction of {Si(n,k ;x)}. Then,f o r i = k ( m - 1) j , 0 < j < k, we have

+

1 Sp'(n, k;X) I where K,

=

(n - 2k

< K,

*

+W2(n

( a - 2K

( a = 0 , 1 ,...,n

/I rmP1 1/(2n-2a-1)'2 2'

+ 2)! -

+ I)! . 2-(2n-2a-1)/2

I f n - 2k

-Fil(n,I)'

l), (5.16.2.1)

( a = 0 , 1,...,n - 1).

(5.16.2.2)

The factor 2--(2n-2ar-1)/2 is omitted at x [F;m(n, k,

-

and

=

a and x [TAm(n,

')

=

b for

- TAl(n>

k)l'

+ 2 < 1 in (5.16.2.2), it is replaced by 1.

Proof. Except possibly at x = a and x = b, we can find points Zi such that Sin-l)(Zi) = 0 and I x - Zi1 &(n- 2k 2)1/2 11 11

<

+

-

5.16.

181

CANONICAL MESH BASES AND THEIR PROPERTIES

by repeated application of Rolle's theorem and the fact that S : k - l ) ( ~ ) vanishes at every mesh point of rmPl . In view of this, it follows that

1 Sy-l)(x)I

< jz 1 Sjn)(x)I 1 dx I < [t(n - 2k + 2 )

*

11 rmPl

(5.16.3)

by Schwarz's inequality. By applying the same argument to Si"-2)(x) and using inequality (5.16.3) rather than Schwarz's inequality, we obtain

1 sy-yx)1

< (n - 2k + 21112 . .(

- 2k

+ 1) . (3)"'" . 11 rm-l

j13/2,

which may be written as

We obtain (5.16.2) by further repetition of this argument. Since the modifications required at x = a and x = b for [Fim- FLJ and [Tdm- T,J are trivial, the lemma is established. We conclude Section 5.16 with Theorem 5.16.2, which is fundamental to Sections 5.17 and 5.18.

Theorem 5.16.2. Let { S , ( x ) } be a canonical mesh basis such that > 0 such that

11 n, I/ = O(1jm). Then there exist real numbers S a ) ( x )=

m

{ S ~ ) ( X )< } ~pa

( a = 0, 1,...,n - 2).

(5.16.5)

i=l

Proof.

We have m


for some positive constants K and

( a = o , l ) ... n - 2 ) )

(5.16.6)

is,.

T h e theorem follows.

REMARK5.16.2. It would be desirable to have (5.16.5) hold for 1 , but inequalities (5.16.2) are not sharp enough. This is reflected later, when we obtain exact replicas of (5.16.2) for generalized splines, indicating that the generality of our methods limits, to a small degree, their sharpness. a =n -

REMARK 5.16.3. I n this discussion, it has been tacitly assumed that d, = r0 is sufficiently large so that the splines Si(n, k ; x ) = S,(X)

v. POLYNOMIAL

182

SPLINES OF ODD DEGREE

+

exist. We also have employed the relation i = k ( m - 1) j, where k ( m - 1) < i km, which was introduced earlier. We henceforth 11 A , I/ = 0" or the assertion that a assume that the statement L'limN+Co basis is a canonical mesh basis for X n ( a , b) or X p n ( a , b) includes II Ti I/ = O ( l / i ) .

<

5.17. Kernels and Integral Representations I n Section 3.16, we determined kernels giving integral representations for the remainders (5.17.1) q-4 =f(4- W f 4; ; now, we determine kernels H,(n, k; x, t ) such that

for functions f(x) in X n ( a , b). If f(x) is in X n ( a , b), it follows that f(x) - Sd(f; x) is in X n ( a , 6 ) ; consequently, (5.17.2) applies to (5.17.1). T h e remainder Pn-l(x) is a polynomial of degree n - 1 to be discussed further at the end of this section.

Theorem 5.17,1, Let f(x) be in X n ( a , b) or X p n ( a , b), and {Si(n, k ; x)} (i = 1, 2 ,...), together with a basis u,(n, K ; x) (i = 1, 2,..., m) for F,(n, k ) be a canonical mesh basis that is an orthonormal basis for X n ( a , b) or X p n ( a , b). Then for every x in [a, b] f ( = ) ( x= ) N-m Zim

s" a

Ha,N(n,K ; x,t ) . f ( " ) ( t )dt

+ f'2;(x)( a = 0, 1,...,n - l), (5.17.3. I)

where Pn-l(x) is a polynomial of degree n - 1, Ha,N(n,k ; x,t ) =

N

m

C uP)(x)

*

u?)(t)

i=l

+ 1 S;")(x) Sj")(t),

(5.17.3.2)

i=1

and the limit exists uniformly with respect to x in [a, b].

Proof. Let {ni}(i = 0, 1,...) be the mesh sequence occurring in the construction of {Si(x)}. By assumption, 11 ni /I + 0 as i + 0 0 ; consequently, f ( - ) ( x )= lim S p ) ( f ;x) N-m

=

lim

N + C C

1

m

i=l

(ui,f)u P ) ( x )

+

N

i=l

i+

(Si,f)Sr)(x)

P,!&(x), (5.17.4)

5.17.

183

KERNELS AND INTEGRAL REPRESENTATIONS

which implies

+

N i=l

S?)(x)

*

+

(5.17.5)

S y ) ( ~ ) I f ( ~dt) ( t P:l,(x), )

where the limits exist uniformly with respect to x by Theorem 5.9.2. T h e theorem is now immediate, since (5.17.5) holds for a = 0, I, ..., n - 1. T h e method of proof for Theorem 5.17.1 depends on the pointwise convergence of S y ) ( f ;x) to f(u)(x).As a consequence, if we restrict ourselves to suitable canonical mesh bases for FA, or P A , , we have the following corollary. * Corollary 5.17.1. Let f(x) be in X Z n ( a ,b) or %:%(a, b), and let (Si(n, k; x)} (i = 1, 2 ,...), together with u,(n, k; x) (i = 1, 2 ,..., m), be a canonical mesh basis for FA,(n, k) or PA,(n, k), which is an orthonormal basis for Z Z n ( a6, ) or &'?(a, b), respectively. Then for every x in [a, b] and a = 0, 1,..., 2n - k - 1, (5.17.3) is valid.

REMARK5.17.1. Observe that H,,,(n, k ; x,t ) is obtained from H,-l,N(n, k; x,t ) by formal term-by-term differentiation with respect to x. I n itself, Theorem 5.17.1 does not establish the existence in L(a, b) of the desired kernels H,(n, k; x, t ) for each x. We show, however, that, for a = 0, 1,..., n - 2, the kernel H,(n, 12; x, t ) exists and is in L2(a,b ) ; it is the limit, in the mean square sense, of the partial sums H a , N ( n , k; x, t). Theorem 5.17.2. Let {ri}(i = 0, I , ...) be a sequence of meshes on [a, b] determining a canonical mesh basis that is an orthonormal basis for Z n ( a , b) or .Xpn(a, b). If H,,N(n, k ; x,t ) is dejined by (5.17.3.21, then, as a function of t , {Ha,N}( a = 0, I ,..., n - 2; N = 1, 2,...) is a Cauchy sequence in L2(a,b) and, consequently, a Cauchy sequence in L(a, b). If Ha(n,k; x,t ) ( a = 0, 1,..., n - 2 ) denotes the common limit, then f ' a ) ( x )=

f Hm(n,K ; x, t ) f ( " ) ( t dt) + P ~ ~ l ( x(a) = 0,1,..., n b

-

2),

(5.17.6)

a

where PnPl(x)is a polynomial of degree n - 1. The convergence is uniform with respect to x in [a, b ] , and H,(n, k;x, t ) is obtained from H,-,(n, k; x,t ) by formal term-by-term differentiation with respect to x.

* For the auxiliary mesh sequence a function of i.

( ~(i3= 0, 1, ...), we

assume Rwt to be bounded as

184

v.

POLYNOMIAL SPLINES OF ODD DEGREE

Proof. Because of the orthonormality of the mesh basis and Lemma 5.10.1, we have

c

N+P

=

1 SI"'(x) 12

i=N+1

'[i = k(m - 1) + j ,

<

0 <j

< k].

(5.17.7)

Consequently, for 01 n - 2, we have a Cauchy sequence, and (5.17.6) follows from (5.17.3); this completes the proof, since the limits exist uniformly with respect to x in [a, b]. We conclude this section with some further comments on mesh bases. I n choosing bases for FA,, Fi, , TA, , and PA,, we can consider them as linear spaces (without identifications) rather than Hilbert spaces. For instance, we can select a basis for FA, including n independent polynomials of degree less than n. These basis elements are identified with the null spline when FA, is interpreted as a Hilbert space. This is consistent, since these polynomials are orthogonal to every element in FA,, including themselves. Their inclusion in the basis allows us to express Syi(f; x) as N

m

and, consequently, to obtain Theorem 5.17.1 and Corollary 5.17.1. Here the polynomials of degree less than n in the basis contribute the term Pgl,(x), and the remainder of the basis %(I.[

k;x), .z(%

k; x),..., .m(% k;4 1

contributes the summation m

1 (f, .I"'(.). Ui)

i=l

Alternatively, we can consider FA, as a Hilbert space with an orthonormal basis ul(n, K ; x), u2(n,k ; x) ,..., u,(n, K ; x). We then define P,-l(x) as

5.18.

REPRESENTATION A N D APPROXIMATION OF LINEAR FUNCTIONALS

185

and (5.17.8) will still be valid.* Both alternatives are in agreement with the formulation of our theorems.

5.18. Representation and Approximation of Linear Functionals I n this section, we examine the representation of linear functionals by means of integrals involving kernels. We consider also the approximation of functionals by other functionals and obtain integral representations for the remainders incurred. We do not limit z :f +f(x). ourselves, as in Section 3.17, to the point functionals 2 T h e representation theorems we establish are of the type established by Peano [1913] and Sard [1963]; their close connection with spline theory was first realized by Schoenberg [1964b]. T h e approach we take is novel but has considerable generality and utilizes heavily the material we have developed here and in Chapter 111. As a consequence of our approach, the theorems we obtain are, in some senses, stronger, and, in other senses, weaker than those obtained by Peano, Sard, and Schoenberg. Letf(x) be in %%(a, b ) or X p n ( a ,b), and let 2 be a linear functional on Z n ( a , b) or .Xpn(a,b) of the form (5.18.1)

where each p j ( t ) is a function of bounded variation on [a,b]. A detailed treatment of functionals of this type can be found in Sard [1963, Chapter f i n the form I]. Let us consider the possibility of representing 90 (5.18.2) where H(t) is a function in L2(a,b) which is independent of f ( x ) and PnPl(x)is a polynomial of degree n - 1. As in the preceding section, we establish first the more general result that there exist sequences {HN(n,k ; t ) } ( N = 1, 2, ...) of functions in %n-k(a, b) such that (5.18.3)

More precisely, we prove the following theorem.

* In (5.17.9), we are thinking of each ui(x) as a particular member of an equivalence class rather than as an equivalence class.

v.

186

POLYNOMIAL SPLINES OF ODD DEGREE

Theorem 5.18.1. Let f ( x ) be in X n ( a , 6 ) or .Xpn(a, b), and let {Si(n,12; x)} (i = 1, 2,...), together with {u,(n, A ; x)} (i = 1, 2 ,..., m ) , be a canonical mesh basis that is an orthonormal basis for X n ( a , b) or X p n ( a ,b). If 9is a linear functional of the form (.5.18.1),then 2of

Zim

=

N-tw

where HN(n,k ; X)

sl

HN(n,k ; t ) - f ( " ) ( t )dt

+ C pi N

m

=

C ai

*

u~)(x)

i=l

3=0

pi =

cs '

l

j=o

a b

a

(5.1 8.4.2)

S:"'(X),

*

(5.18.4.1)

i=l

uaj)(s)dpj(s)

mi =

+ 20PnP1,

Sij)(s)dpj(s)

(i = 1 , 2,..., m),

(5.18.4.3)

1, 2,...,N ) ,

(5.18.4.4)

(i

=

and PnP1(x)is a polynomial of degree n - 1.

Proof. We can represent f ( a ) ( x )( a = 0, 1, ..., n

j 1 (f, m

f ( a ) ( x ) = lim

ui)up)(x>

N+ w

i=l

+ c (f,SJ

-

1)

as

+p~2~(x),

N

s?)(X)j

i=l

where the limit exists uniformly with respect to x; hence,

Since the convergence is uniform with respect to x, 2of

=

lim

N-tm .

j

J~ f (j,ui)uij)(s)+ a

i=1

However, by definition we have

c (j,si)sij)(s)i dpj(s) + N

i=l

2 0

P,-~

.

dt;

(5.18.5)

consequently, after substituting for (f, ui) and the order of summation, we obtain

(f,8,) and changing

(f,g) = J

from which the theorem follows.

h t ) g'"'(t)

5.18.

REPRESENTATION AND APPROXIMATION OF LINEAR FUNCTIONALS

187

Corollary 5.18.1. Let f ( x ) be in X 2 n ( a , b) or X:"(a, b), and let {Si(n,k; x)) (i = 1, 2,...), together with {u+(n,k; x)) (i = 1, 2,..., m), be a canonical mesh basis that is an orthornomal basis f o r Z n ( a , b) or Z p n ( a , b). I f 9 is a linear functional of the f o r m

e 2% - 2),

(7

(5.1 8.6)

where each p j ( t ) is a function of bounded variation on [a, b], then (5.18.4) holds except that 7 2n - 2.

<

For functionals of the form 64of

S B f ( j ) ( t )&(t)

=

a

j-0

(7

< n - 2),

(5.18.7)

where each p j ( t ) is a function of bounded variation on [a, b], we obtain representations of the special form (5.18.2).

Theorem 5.182. Let {ri}(i = 0, I, ...) be a sequence of meshes on [a, b] determining a canonical mesh basis that is an orthonormal basis for %%(a, b ) or %-"(a, 6). I f HN(n,k; x) is dejined by (5.18.4) except that 7 n - 2, then {H,(n, k; x)> ( N = 1, 2, ...) is a Cauchy sequence in L2(a,b) and, consequently, a Cauchy sequence in L ( a , b). I f H(n, k ; x) denotes the common limit, then

<

640f

b

=

a

H(n, k ; t ) f ( " ) ( t dt )

+ 640 P,-l

,

(5.18.8)

where Pn-l(x) is a polynomial of degree n - 1. Proof. The proof of Theorem 5.17.2 applies, essentially unchanged. We are now in position to consider the approximation of a linear functional 2 0 f the form (5.18.1) by linear functionals of the special form r

B of

=

k-1

C C auf(j)(xi) i-0 j=o

(k

< n),

(5.18.9)

determined by a set of constants aij (i = 0, 1,..., r ; j = 0, 1,..., k - 1) and a mesh A : a = xo < < x, = b. If we let p j ( t ) be a step function with jumps aij at xi (i = 0, 1,..., r ) , then (5.18.9) takes the form B of

=

yp j-0

a

( t ) d&(t),

(518.10)

188

v.

POLYNOMIAL SPLINES OF ODD DEGREE

where each pi(t) is a function of bounded variation on [a, b ] ; thus, our representation theorems apply. Let pl(n, k; x), u2(n,k;x), ..., u,(n, k ; x) be an orthonormal basis for FA(n, k ) , and let {Si(n,k; x)) (i = 1, 2, ...) extend this basis to an orthonormal basis for %"(a, b). We could equally well consider TA(n, k) or, in the case of 3fpn(a,b), f',(n, k); but Fi(n, k) encounters difficulties at x = a and x = b. We now make two important observations. T h e first observation is that, since Sp)(xj) = 0 ( a = 0, 1,..., k - I ; j = 0, I ,..., Y ; i = 1, 2,.,.), the coefficients Pi(i = 1, 2, ..., N ) , defined by (5.18.4.4), all vanish for B. This occurs since all the mass of the measures p i ( t ) is concentrated at the mesh points of A ; moreover, this is true for every N . Consequently, if we apply either Theorem 5.18.1 or Theorem 5.18.2 to 9 - B, the coefficients Pi for 2 - B are determined by 9 alone and are unaffected by B.Our second observation is that

90Sd,f =

Jl]5

aiuy)(n,k; t ) .f'")(t)dt

i=l

I+

90 P,-l

(5.18.11)

so that

as a re-examination of the proof of Theorem 5.18.1 reveals. Since we can determine the coefficients aii in (5.18.9) such that (5.18.13)

Bof =~ o S A , ~ ,

we have the following two theorems.

Theorem 5.18.3. If 2 is a linear functional de$ned by (5.18.1) and is the linear functional defined by (5.18.13), then

if B

90 f - B of

=

kz 1:

1

N i=l

p i . S p ) ( n ,k ; t ) - f ( " ) ( t d) t l ,

(5.18.14.1)

where "

6

pi = C Ja Slj)(n, k;S) dpj(s) j-0

(7

< n)

(5.18.14.2)

and the pj(s) ( j = 0, 1, ..., 7) are functions of bounded variation on [a, b] determined by 9alone.

5.18.

REPRESENTATION AND APPROXIMATION OF LINEAR FUNCTIONALS

189

Theorem 5.18.4. If 9is a linear functional of the form (5.18.7) and B is a linear functional of the form (5.18.13), then H ( n , k; t ) f ' " ' ( t ) dt,

(5.18.15.1)

where H(n, k; t ) is in L2(a,b) and is the limit, in the mean square sense, of HN(n,k; t ) =

m

olp?)(n, i-1

k ;t )

+

N

BiSln)(n,k; t ) ,

(5.18.15.2)

i=l

(5.18.15.3) (5.18.15.4)

The pj(s) are functions of bounded variation on [a, b] and depend on 9 alone, being given by the representation (5.18.7) f o r 9.If B is any linear functional of the form (5.18.9) and 2'- B annihilates polynomials of degree less than n, then 9 0 f -B o f

=

J: H(n, k ; t ) f ' " ' ( t ) dt,

(5.18.16)

where H(n, k ; t ) is in L2(a, b) and is dejined as in Theorem 5.18.2. Moreover, (5.18.17)

is minimized when B is of the form (5.18.13).

T h e application of the methods of this section to the point functionals --+ f ( x ) is illuminating, but we defer further discussion of these functionals until Chapter VI. There we also examine other functionals of considerable interest. For instance, if we do not wish the approximating functional B to involve the value of f(x) at xi , we cannot merely set ai, = 0 in (5.18.9) and, in general, satisfy (5.18.13). Approximating functionals exist, however, which meet these requirements. This is a desirable property in obtaining predictor or corrector formulas for numerical integration, since stability requirements often require the suppression of certain values of the function and its derivatives in these formulas.

zz: f

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CHAPTER VI

Generalized Splines

6.1. Introduction

T h e theory of polynomial splines of odd degree can be approached from two points of view: (1) the algebraic, which depends primarily on the examination in detail of the linear system of equations defining the spline; or (2) the intrinsic, which exploits the consequences of basic integral relations existing between functions in X n ( u , b), or X 2 n ( u , b), and approximating spline functions. T h e first approach reaches its high point for the cubic and doubly cubic spline but ebbs noticeably even for polynomial splines of higher odd degree. For generalized splines, there is at present no algebraic theory, although special theories for particular generalized splines could well evolve. T h e intrinsic approach, however, carries over undiminished and reaches its fullest maturity in the generalized setting. I n Chapter VI, we develop this theory as a continuation of the intrinsic theory of polynomial splines. I n general, we assume that we have a linear differential operator L defined by L

E

a,(.)

*

D" + U"&)

*

D"-1

+ ... + a,(.),

(6.1.1)

where each u j ( x ) ( j = 0, 1,..., n) is in Cn[u,b], and un(x) does not vanish on [a, b]. Let L* be the formal adjoint of L ; thus, L*

= (-1)"

+

***

*

D"{U"(.)

-

-}

+ (-1y-1 - D"-l{un&)

*}

(6.1.2) W,(.).I + a&). < xN = b is a mesh on [a, b], then a generalized

If 4:u = x,, < x1 < spline of deficiency k (0 k n) with respect to d is a function S,(X) which is in X2n--k(a,b) and satisfies the differential equation

< <

L"LSA

=0

(6.1.3)

on each open mesh interval of A . We also say that SA(x)has order 2n when we want to indicate the order of the operator L*L defining S,(X). 191

192

VI.

GENERALIZED SPLINES

If K = 0 and L has analytic coefficients, then Sd(x) has continuous derivatives of all orders and satisfies (6.1.3) throughout [a, b ] ; in this case, continuity of the (2n - 1)th derivative implies continuity of the 2nth and all higher derivatives. Thus, for this important class of differential operators, k = 0 is equivalent to continuity of all derivatives. T h e ordinary spline (deficiency one) allows discontinuities in the (2n - 1)th derivative, but only at mesh points. I n general, the deficiency of a spline is a measure of the failure of the spline to satisfy (6.1.3) on [a, 61.

6.2. The Fundamental Identity If L is a differential operator of order n and L* is its formal adjoint, then we have the identity Lu(x) . W ( X )

+ U(X) . L*v(x);

= (d/dx) P[u(x),a(.)]

(6.2.1.1)

here P[u(x),~ ( x ) ]is the bilinear concomitant and is defined by

cc

n-1 n-j-1

P[u(x),v ( x ) ] =

j=O

(-l)k

. u(n-j-k-1)

(x)

. {u,+(x)

*

w(x)}(".

(6.2.1.2)

k=O

T h e right-hand side of (6.2.1.2) can be regrouped so that

P[u(x),ZJ(X)] =

1dn-j-')(x) C (-1)' *

. { u ~ - ~ - ~ ( x~(x)}'"'. ) (6.2.2) *

k=O

3=0

< xN = b be given, and let S,(x) Now let A : a = xo < x1 < be a generalized spline on A of order 2n. If in Eqs. (6.2.1) we set u ( x ) = f ( x ) - S,(x) and v ( x ) = LS,(x), then we obtain, after integration over the interval [xiPl,x i ] , the relation

c1

L{f(x)

=

-

] "c').(fi j=O

Sd(XH

LSd(X) dx

c( j

- S,(x)}(n-j-1)

f

k=O

-1 ) k

*

{an--)&)

*

-qj(X)}(k)

1 IL

(i = 1, 2,..., N), (6.2.3) since L*LS,(x) = 0 on the open mesh interval (xi-l, xi). Consequently, in view of the general identity

=

Sb a

( L ~ ( Xdx ) }~2

L { f ( x ) - S,(x)} a

. LS,(x) dx

-

1:

{ L S , ( X ) }dx, ~

(6.2.4)

6.3.

193

THE FIRST INTEGRAL RELATION

it follows from (6.2.3) that

J",(."

- s,(x)1}2 dx

T h e identity (6.2.5) is the fundamental identity for generalized splines; the only restrictions are (1) S,(x) is a spline with respect to A , and 1 (2)f ( x ) is in X n ( a , b).

6.3. The First Integral Relation Under certain restrictions on S,(x), the fundamental identity reduces to

which we call the jirst integral relation for generalized splines. If S,(x) is of deficiency K , then { L S , ( X ) } ( ~( -j ~=) 0, I, ..., k) normally will have discontinuities at the mesh points of A . Let 7

P,(W; X) = 1 (-l)i{~~-~-,(~)La(x))l') (r = 0, 1,...,

- 1);

(6.3.2)

i=O

then P,(S, ; x) is the coefficient of {f(x) - S,(X)}(~-~-~) in (6.2.3). Since we do not require ,k$(S, ; x) to be continuous at mesh points for r >, n - k, we compensate for any lack of continuity by requiring that f ( ' ) ( x ) - ST)(x) ( a = 0, I , ..., K - 1) vanish at the interior mesh points of A ; if, in addition, we impose suitable end conditions at x = a and x = b, then (6.3.1) is valid. One acceptable set of end conditions is that f(x) - S,(x) vanish at x = a and x = b and f ( x ) - S,(x) be of type 1'; another is that f ( x ) - S,(x) vanish at x = a and x = b and that Sd(x)be of type 11'. We emphasize here that the concepts "type I"' and "type 11"' depend on the underlying operator L. We say that a function f(x) is of type I' provided f ( " ) ( x )vanishes at x = a and x = b for a = 1, 2, ..., n - 1, where n is the order of the operator L. On the other hand, we now say

194

VI.

GENERALIZED SPLINES

that a function f(x) is of type 11’provided {Lf(x)}(m)vanishes at x = a and x = b for a = 0, 1,..., n - 2. For L = Dn, this agrees with the earlier definition. * Equivalence classes, etc., can be introduced as in Chapters I11 and V. As before, a spline is simp2e if it is in C2n-2[a,b]. For generalized splines of interpolation of type k, this condition is also sufficient; a generalized spline SA(f ; x) of deficiency k on A is a spline of interpolation of type k ;f S y ) ( f ;x) ( a = 0, 1,..., k - I ) interpolates ~ )0 to the values of f(”(x) at the mesh points of A and { L S A ( X ) } (= ( a = 0 , 1,..., n - k - 1) at x = a and x = b. I n the periodic case we require that f(a)(x)- Sp)(f;x) ( a = 0, 1,..., k - 1) vanish on all of A and S,(f; x) be in Czn-k-l[a, 61. Theorem 6.3.1 summarizes these conditions under which we have indicated the validity of the first integral relation (6.3.1).

Theorem 6.3.1. Let A : a = xo < x1 < < xN = b and f(x) in .%?(a, b) be given. If SA(f;x) is a generalized spline of deficiency k on A

such t h a t f ( . ) ( x ) - S$)(f;x) ( a = 0, 1,..., k - 1) vanishes at the interior mesh points of A and, in addition, one of the conditions, (a) f(x) - S,(f; x) isof typeI’, andf(xi) - Sd(f;xi) = O(i = 0, N ) , (b)f(xi) - S A ( f ;xi) = 0 ( i = 0, N ) and S,(f; x) is of type II’, (c) S,(f; x) is of type k , (d) f(a)(xi) - S y ) ( f ;xi) = 0 ( a = 0, 1,..., k - 1; i = 0, 1,..., N ) f(x) is in X p n ( a , b), and SA(f ; x) is periodic, is satisjied, then the first integral relation (6.3.1) is valid. T he first integral relation is valid for another broad class of generalized splines, besides the classes covered by Theorem 6.3.1. These splines have the property that some of their first n - 1 derivatives are prescribed at mesh points. At a given mesh point, the specified derivatives need not be consecutive, as in the case of splines of deficiency K , and at different mesh points different derivatives may be prescribed. For example, the even derivatives may be specified at alternate mesh points and odd derivatives at the remaining mesh points; we include f(x) among the even derivatives. I n addition, we require Sy)(x) ( a = 0, 1,..., n - 1) to = f(j)(xi),/3,++,(Sd ; x) need be continuous. However, when Sij)(xi) not be continuous at xi, but when S$)(x,) # f(j)(xi), we require (6.3.3) Finally, if S y ) ( x )# f(j)(xi) at x = a or x = b, then /3n-j-l(SA; x) must vanish there. We refer to splines determined by such conditions as heterogeneous splines. For heterogeneous splines, the following theorem is a direct consequence of the fundamental identity.

* This is equivalent to

p m ( f ;a) = &(f; b )

=

0 (m

=

0,1 , ..., n

-

2).

6.4.

THE MINIMUM NORM PROPERTY

195

Theorem 6.3.2. Let d: a = x,, < x1 < *.- < xN = b and f ( x ) in &-n(a, b) be given. If S,(f;x ) is a heterogeneous spline on d such that, when s(=)(f; xi) ( a = 0, 1,..., n - 1; i = 0, 1,..., N ) is prescribed, we have ST'(f;xi) = f(=)(xi),then

6.4, The Minimum Norm Property Under the hypotheses of either Theorem 6.3.1 or Theorem 6.3.2, we know from the first integral relation that

We carry over to the setting of generalized splines the terminology of polynomial splines and refer to this extremal property as the minimum norm property; for L = Dn, inequality (6.4.1) expresses the minimum norm property for polynomial splines, as it should.

Theorem 6.4.1. Let A : a = x,, < x1 < * * - < X , = b and Y = {yie I i = 0, 1,..., N ; 01 = 0, 1,..., k - l} be given. Then of all functions f ( x ) in 'X"(a, 6 ) such that f ( = ) ( x i= ) yi= (i = 0, 1,..., N ; a = 0, 1,..., k - I), the generalized spline Sd(Y ;x ) of type 12, when it exists, minimizes (6.4.2)

If g(x) also minimizes (6.4.2),then g(x) and S,( Y ;x ) differ by a solution of Lf = 0. Moreover, ifyo== yN=( a = 0, 1,..., k - l), then of all functions in 'Xpn(a,b), the periodic generalized spline of type k, if it exists, minimizes (6.4.2) andis unique in this sense to within a periodic solution of L f = 0. For k = 1 and L E D", Theorem 6.4.1 reduces to the main assertions of Theorem 5.4.1. Although we could formulate the theorem in terms of the other conditions under which Theorem 6.3.1 asserts the validity of the first integral relation, we formulate it only for heterogeneous splines and consider these alternative theorems as special cases of the latter.

Y

Theorem 6.4.2. Let A: a = xo < x1 < -.. < xN = b and = {yioLi) be given, where i ranges over a subset of 0, 1,..., N and airanges

196

VI.

GENERALIZED SPLINES

over a subset of 0, 1, ..., n - 1 which varies with i. Then of all functions f ( x ) in X n ( a , b) such that f("*)(xi)= yin, when yiu,is in Y , the heterogeneous generalized spline S,( Y ;x) for which Y is suficient to define S,( Y ;x), if it exists, minimizes

J; {Lf(41dx. 2

(6.4.2)

If g ( x ) satisfies the required conditions and also minimizes (6.4.2), then g ( x ) and S,( Y ;x) differ by a solution of Lf = 0. Generalized splines do not exist for all operators L on every mesh; ordinarily, when there are sufficiently many mesh points they do exist. We cannot, however, be as explicit with respect to the number of mesh points required for existence as we can for polynomial splines. For this reason, we have explicitly required the existence of the splines S,(f; x) in the statement of Theorems 6.4.1 and 6.4.2. For polynomial splines, the mesh was required to have sufficient points so that the splines involved exist. This question is discussed further in Section 6.7.

6.5. Uniqueness T h e methods of Chapters I11 and V carry over directly and yield the expected uniqueness theorems. We need only observe that, if S,( Y ; x) and s,(Y;x) are two generalized splines interpolating to a common interpolation vector Y on A , then their difference S,( Y ;x) Y ; x) is a generalized spline on A of the same type, but its interpolation vector is the zero vector. I n addition, S,(Y; x) - s,(Y;x) has the same continuity properties at mesh points which S( Y ;x) and s,( Y ;x) have. We can now argue, using the minimum norm property, that S,( Y ;x) - S,( Y ;x) is the zero function Z(x). This argument establishes both Theorem 6.5.1 and Theorem 6.5.2, which follow.

sA(

Theorem 6.5.1, Let A : a = xo < x1 < < xN = b and Y = {riaI i = 0, 1,..., N ; CY = 0, 1,..., k - l } be given. In addition, let L and A be such that, if L g = 0 and g(a)(xi)= 0 (i = 0, 1, ..., N ; CY = 0, 1, ..., k - I), then g ( x ) = 0. Under these conditions, there is at most one generalized spline S,( Y ;x) of type k on A such that Sy)(Y ; xi) = yiDI(i = 0, 1,..., N ; CY = 0, 1,..., k - 1); the periodic generalized spline of deficiency k is also unique under these hypotheses. Y

Theorem 6.5.2. Let A: a = xo < x1 < < xN = b and 5 {yiai> be given, where i ranges over a subset of 0, 1,. .., N and airanges

6.6.

197

DEFINING EQUATIONS

over a subset of 0, 1, ..., n - 1 which varies with i. I n addition, let L and A be such that, i f Lg = 0 and g("l)(x,)= 0 for all allowable values of i and 0 1 ~, then g(x) = 0. Under these conditions, there is at most one heterogeneous spline SA(Y ;x ) such that Sk%'( Y ; xi) = yiu,for all allowable values of i and ori and for which Y is suflcient to define Sd( Y ;x). 6.6. Defining Equations Let uI(x),uz(x),..., uBn(x)be a fundamental set of solutions of the differential equation L*Lf = 0 on [a, b] such that u(j.)(a)= Si,u+l ( j = 1 , 2,..., 2n; 01 = 0, 1,..., 2n - 1). If a mesh A :

< <

is given, then in each mesh interval xiPl x xi (i = I , 2,..., N ) a generalized spline SA(x)has a unique representation

S,(X)

zn

=

1

C{jUj(X).

(6.6.1)

j=1

Interpolation requirements, continuity requirements, and end conditions determine a system of 2nN linear equations for obtaining the 2 n N coefficients cii . We defer until the next section the question of the existence of a solution to these equations and content ourselves in this section with the task of formulating the appropriate equations for a number of important cases. T h e appropriate equations for periodic and type I generalized splines of deficiency k can be obtained from Eqs. (5.7.4) and (5.7.3) by the following modifications:

(1) Equation (5.7.4.1) must be evaluated at xi rather than at hi . (2) Equation (5.7.3.7) must be evaluated at xN rather than at h, . (3) Equation (5.7.4.2) must be replaced by

VI.

198

GENERALIZED SPLINES

In the case of generalized splines of type 11, in addition to modifications (1) and (3), we must also have

(6.6.2.2)

and

Cft, =

(6.6.2.3)

instead of (5.7.3.6) and (5.7.3.8). For generalized splines of type R , the situation is the same as for type I1 splines except that (5.6.2.2) and (5.6.2.3) are replaced by

c:, =

(6.6.2.4)

c:, =

(6.6.2.5)

and

6.7.

EXISTENCE

199

I n the preceding equations, the notation { L V ) signifies ~) the crth derivative of Lv evaluated at x = c. T h e fact that the linear differential operator L does not, in general, have constant coefficients forces us to evaluate the uj(x) at xi rather than at h i , since in the general case uj(x - a ) is not a solution of Lf = 0 even though q(x) is a solution. Heterogeneous splines have no systematic pattern and have to be handled individually. One should observe, however, that at each interior mesh point S y ) ( x ) is continuous for 01 = 0, 1,..., n - 1. I n addition, either S y ) ( x )is specified or &(S, ; x) is required to be continuous for 01 = 0, 1,..., n - 1 and 01 y = n - 1. T h u s we have 2n(N - 1) linear equations from these conditions. At x = a and x = b, either Sp)(x) is prescribed or &(SA ; x) is required to vanish for 01 = 0, 1,..., n - 1 and 01 y = n - 1 . This gives us 2n additional equations or a total of 2nN equations; there are exactly the same number of coefficients cii to be determined. I n formulating the system of equations appropriate to a heterogeneous spline, one should note that &(S, ; x) operates on SA in a linear fashion. Thus,

+

+

6.7. Existence

We are now in position to apply the analysis of the preceding three sections to establishing the existence of generalized splines. T h e proof proceeds along the same lines as the existence proofs of Chapters I11 and V.

Theorem 6.7.1. Let A : a = x, < x1 < < x N = b and Y = (yim1 i = 0, 1,..., N ; 01 = 0, 1,..., k - I} be given. I n addition, let L and d be such that, ;f Lg = 0 and g(.)(x,) = 0 ( i = 0, 1,..., N ; a = 0, 1,..., k - l), theng(x) = 0. Under these conditions, thegeneralized spline S,( Y ;x) of type k on d such that S y ) (Y ;xi) = y i ( i = 0, 1, ..., N ; 01 = 0, 1,..., k - 1) exists; the periodic generalized spline of type k S,( Y ; x ) also exists. Proof, We know that, if SA(Y;x) exists, it is unique by Theorem 6.4.1. Moreover, from Section 6.6 we know that we can obtain a system of linear equations for determining S,(x) of the form AC

=

Y,

(6.7.1)

where the components of the vector C are the coefficients cij , A is a matrix, and Y is a vector. Since two distinct sets of cij define two distinct

200

VI.

GENERALIZED SPLINES

splines, A C = 0 has a unique solution; therefore, A-l exists. T h e theorem follows. Similarly, we can establish the following existence theorem for heterogeneous splines.

Theorem 6.7.2. Let A : a = xo < x1 < .--< xN = b and Y = (yibt) be given, where i ranges over a subset of 0, 1, ..., N and cii ranges over a subset of 0, 1,..., n - 1 which varies with i. In addition, let L and A be such that, ;f Lg = 0 and g("i)(xi) = 0 f o r all allowable values of i and mi , then g ( x ) = 0. Under these conditions, the heterogeneous spline S,( Y ; x ) , such that S 2 ) (Y ;xi) = ytetf o r all allowable values of i and cii and Y is suficient to define S,( Y ; x), exists. T h e existence arguments we have used depend on first establishing uniqueness. T h e question of uniqueness, in turn, centers on the question of how often a solution f of L*Ly = 0 and its derivatives can vanish on [a, b] without vanishing identically. For a specific differential operator L such as L = Dn,this question usually has a very precise answer. For the general case, the situation is somewhat less precise. We have, however, two basic results whose easy proofs we omit. First, for the operators L under consideration, there is a maximum number of times a solution of L*Ly = 0 and certain of its derivatives can vanish without vanishing identically. T h e maximum number, unfortunately, is not explicitly given. T h e second result is that, in any sufficiently small interval, 2n independent zero interpolation requirements on y and certain of its derivatives force y ( x ) to vanish identically; the maximum length of such an interval is again elusive.

p. dc 1

6.8. Best Approximation

+\ W' l : "

Both generalized splines of type k and heterogeneous splines depend linearly on their defining values. Th u s we have in both cases S,(f - g ; ).

= S,(h x) - S,(g;

4.

< xN = b and f ( x ) in X n ( a , 6 ) be given. Let A : a = xo < x1 < If S,(x) is any spline on A , we have S,(f - Sd ;).

4 - S,(SA

=

S,(h

=

S,(h ). - Sd(X),

;

provided S d ( f - S, ; x), S,(f; x), S,(S, ; x) have their defining values determined by f ( x ) - S,(x), f(x), and SA(x),respectively; continuity

6.9.

CONVERGENCE OF LOWER-ORDER

DERIVATIVES

20 1

requirements must also be compatible, however. I t now follows from the Minimum Norm Property that

1I.f - ' A 1 '

- 11

SA,f-S~

11'

=

1I.f - ' A

- SA,f-SA

/I2

< xN = b and A x ) in Theorem 6.8.1. Let A: a = xo < x1 < X n ( a , b) be given. Then of all generalized splines SA(x) of type k on A , the spline of interpolation S,( f ; x) with S( ") ( f; xi) = f ( " ) ( x i ) (i = 0, 1,..., N ; 01 = 0, 1,..., k - l), i f it exists, minimizes

1:

{ L j ( x )- L s A ( x ) } 2

dx.

(6.8.1)

I f s A ( x )also minimizes (4.8.1), then S,(X)and s A ( f ; x ) differ by asohtion of Ly = 0. Furthermore, if f ( x ) and S A ( x ) are required to be periodic, the spline of interpolation of type k again minimizes (6.8.1) and is unique in this sense up to a periodic solution of Ly = 0. We have, in addition, the following theorem.

Theorem 6.8.2, Let A : a = xo < x1 < *.- < xN = b and f ( x ) in X n ( a , b ) be given. Then of allgeneralized heterogeneous splines S,(X)on A , determined by a specified set of continuity conditions and defining values S3'(xi), where i ranges over a subset of 0, 1, ..., N and oli ranges over a subset of 0, I , ..., n - 1 which varies with i, the spline of interpolation s A ( f ; x ) with S$l)(f;xi) = f'"t)(xi) for allowable values of i and ai and f o r which the interpolation requirements are sufficient to define sA( f ; x ) , i f it exists, minimizes (4.8.1).If S A ( x )also minimizes (6.8.1), then S,(X)and s A ( f ; x ) differ by a solution of Ly = 0. 6.9. Convergence of Lower-Order Derivatives I n discussing convergence, we confine ourselves to generalized splines of type k ; we point out, however, that the periodic generalized spline

202

VI.

GENERALIZED SPLINES

of type k, the type I generalized spline of deficiency k, and the type I1 generalized spline of deficiency k also have regularly occurring interpolation points to an approximated function f ( x ) and its derivatives, so that Rolle's theorem can be applied to them in the same manner as it is to polynomial splines in Section 5.9. Thus, analogous convergence properties are obtainable. We formulate our theorems to include all these splines, but for the most part the proofs and discussions apply explicitly only to the case of type k splines. T h e heterogeneous splines that we consider have the irregularities in their interpolation and continuity properties confined to the initial mesh of a given mesh sequence {A,} ( N = 1, 2 ,...), and we assume that A , C A,(N = 1 , 2,...). Simplicity is gained by requiring that SdN(f; x) = f ( x ) at the mesh x) be of deficiency 1 (Section 1.2) points of A , not in A , and that SdN(f; at these points. T h e pattern of the convergence argument in this section is that of Section 5.9, with one important change involving an application of Minkowski's inequality. x)] ( N = 1, 2, ...) be a sequence Letf(x) be in Z n ( a , b), and let {SdN(f; of generalized splines of type k . If { A , : a = xoN < xIN < < ,:x = b} is the associated sequence of meshes, then SYA(f;xiN)= f(")(xiN) (i = 0, 1,..., mN ; a = 0, 1,..., k - 1). By repeated application of Rolle's theorem, we know that SyJ(f; x) interpolates to f(a)(x) at least once in every OL - 2k 2 consecutive mesh intervals for 01 2k and at least once in every mesh interval for 0 a 2k - 1. Consequently, for each x in [a, b] we can find an xu, such that SyJ(f; xu,) = f ' & ) ( x O N ) (0 a n) and 1 x - x,, I K, 11 d, 11, where K, = a - 2k 2 if 2k a n - 1 and K, = 1 if 0 a min (2k, n - 1 ) . Furthermore, if x # a or x # b and N is sufficiently large so that there are points of interpolation both to the left and right of x,then Ku = ( a - 2k 3)/2 if 2k a n - 1. It follows that

+

< <

< < < <

<

+

< <

+

< <

JZ =

j' { f ( " ) ( x ) SF;( f;x ) } ~dx. -

(6.9.1.2)

a

Here, as in earlier chapters, we have made use of Schwarz's inequality. We repeat this process for 01 = n - 2 but employ Eqs. (6.9.1) rather than Schwarz's inequality; thus, if(n-2yX)

-

sl;",-zyf; x)l G J

K ~ . 11 ~A , ,11312.

{ ~ ~ - ~ } 1 a/ 2

(6.9.2)

6.9.

CONVERGENCE OF LOWER-ORDER DERIVATIVES

203

T h e general case is described by I f ( " ) ( x )- SlpA(f;%)I

<J

*

{Kn-l}'/z Kn-z ... . K, /I A , (a = 0,

1/(Zn-2m-1)/2

I ,..., 71 - 1). (6.9.3)

We can now use (6.9.3) to establish a bound for J , which in turn can replace J in (6.9.3); this will establish the uniform convergence of S$i(f; x) tof'")(x) (a = 0, 1,..., n - 1) for x in [a, b]. We have by definition W ( x ) - S A , ( ~x)> ;

+ an-,(x> D"-'(f(x)

an(%)W f ( x ) - S A J A x)> -

SdN(f; x>>

+ ..* + ao(x)(f(x>

-

S A J ~x>>, ; (6.9-4)

by Minkowski's inequality. If we transpose terms and make use of (6.9.3), we obtain K-' . J

<

1s"{Lf(x)

-LSA,(f; x)}z d x /

112

(6.9.6.1)

where

. {Kn-'}'I2 Kn+ *

* *.*

K3 // A ,

jl(2n-z3--1)/2].

(6.9.6.2)

Since a,(x) does not vanish and ai(x) ( j = 0, 1,..., n) is continuous on [a, bJ, for 11 A , JJ sufficiently small K is positive and bounded. Thus, if either the minimum norm property holds or the best approximation property holds with A , C A , ( N 3 l), then J is bounded. This implies Theorems 6.9.1 and 6.9.2, which follow.

Theorem 6.9.1. { A , :a

Let f(x) in X n ( a , b) and

= xoN

< xlN <

*-*

< x;

N

= b}

(N

=

1,2,...)

with 11 A , (1 4 0 as N -+ co be given. Let (SdN(f; x)) ( N = I , 2, ...) be a sequence of generalized splines of deficiency k where sA,( f ; x) is a sphW on A , with SF$(f;xi") = f ( a ) ( x i N ) (a = 0, 1, ..., k - 1; i = 1, 23...3 mN - 1)

204

VI.

GENERALIZED SPLINES

such that one of the conditions, (a) each SdN(f;x) is of type K , (b) f ( x ) is in X p n ( a , b), StJf; xiN) = f ( a ) ( x i N )( a = 0, 1, ..., k - 1; i = 0), and each S d N ( fx) ; is periodic, ( c ) f ( x ) - S d N ( fx) ; vanishes at x = a and x = b and is of type I' ( N = 1, 2, ...), (d)f(x) - S,,(f; x) vanishes at x = a and x = b, A , C A , ( N 3 l), and Sdl(f;x) - SdN(f; x) is of type 11' ( N = 1, 2,...), is satisfied, then S$$(f;x) converges uniformly to f(=)(x)on [a, b] f o r a = 0, 1 ,..., n - 1 and If(,=)(x) -

sy(f;x)l = O(1l A,

1)(2n-2=-1)'2

1-

(6.9.7)

Theorem 6.9.2. Let f(x) in X n ( a , b ) and {A, :a

= xoN

< x I N < ... < x,"N

=

b} ( N

=

1, 2, ...)

with I] A , jl -+0 as N -+ co and A , 5 A , ( N 2 1) begiven. Let {S,,(f;x)> ( N = 1, 2,...) be a sequence of generalized heterogeneous splines where SAN( f ; x) is a spline on A , which has dejiciency one at each mesh point not in A , and interpolates to f ( x ) at these points. In addition, for i ranging over a subset of 0, 1,..., m1 and airanging over a subset of 0, 1,..., n - 1 which varies with i but is independent of N , let Sy;'(f; xi1) = f(a*)(x:), and let these interpolation requirements be sufficient to define each SdN(f;x). Then SY;)(f; x) converges uniformly t o f ( = ) ( xon ) [a, b]f o r 01 = 0, 1 , ..., n - 1, and (6.9.7) applies.

6,IO. The Second Integral Relation As when we deduced the first integral relation, our point of departure is the basic identity {Lu(x)}* ~ ( x = ) (d/dx) P[u(x),~(x)]

+ U(X) - (L*V(X)}

(6.10.1)

existing between a linear differential operator L and its formal adjoint L*. Let A : a = xo < x1 < < xN = b and f(x) in 3?2n(a, b) be given. If we set u(x) = f ( x ) - S,(x) and v(x) = L f ( x ) - LS,(x), where Sd(x) is a generalized spline on A associated with the operator L, then

6.10.

205

THE SECOND INTEGRAL RELATION

If we let

p,(f

c T

- SA ;3) =

(-1p[@n-,-j(x)

.(Lf(x)-LSA(x)}]"'

j=O

( r = 0, 1,..., 71

- l), (6.10.3)

then we can express P[j(x)- SA(x),Lf(x)- LS4(x)]as P[f(X) - s A ( x ) , Lf(x)- LsA(x)] =

c { f(x)

n-1

-

SA(x)}(n-j-l)

*

pj(f - sA

; .)* (6.10.4)

j=O

T h e relation (6.10.4) follows from (6.2.2), and the notation (6.10.4) is consistent with the notation in Section 6.3. As a consequence of (6.10.2) and (6.10.4), we obtain the identity

When the summation

vanishes, we obtain

this is the second integral relation for generalized splines. If in the (Y < n) at summation (6.10.6) we have f(=)(xi) = Sy)(xi) for a (0 an interior mesh point xi , then there is no contribution from

<

{f(x)

- SA(X)}(*) . /%-*-I(

f - SA ; X)

at x p to the summation (6.10.6).On the other hand, iff(.)(xi) # %'(Xi), we require lim Lap1(f - SA ; x) = lim &-a-l(f - S A ;x) (6.10.8) x+x,+

X+Zz-

in order that there be no contribution to the summation. At either = a or x = b if f(=)(x)- Sy)(x) does not vanish, we require pn--l(f- SA ; X) to vanish at the point in question to avoid a contribution. Theorems 6.10.1 and 6.10.2 now follow.

x

206

VI.

GENERALIZED SPLINES

Theorem 6.10.1. Let f ( x ) in Z Z n ( a ,b) and A : a

= x,,

< x1 < ... < xN = b

be given. Let Sd(f;x) be a generalized spline on A of deficiency k with S y ) ( f ;xi) = f ( a ) ( x , )( a = 0, 1,..., k - I ; i = I , 2,..., N - 1 ) such that one of the conditions, (a) S t ) ( f ;xi) = f ( a ) ( x , () a = 0, I , ..., k - 1 ; i = 0, N ) and {Lf(xi) - L S d ( f ;xi)}(*) = 0 ( a = 0, 1,..., n - k - 1 ; i = 0, N ) , ( b ) f ( x ) is in %:"(a, b), Sd(f;x) is periodic, and S$)(f;xi) = f ( a ) ( ~ i ) ( a = 0, 1,..., K - 1 ; i = 0), ( c ) f ( x ) - S,(f; x) vanishes at x = a and x = b and is of type If, (d) f ( x ) - S A ( f ; x) vanishes at x = a and x = b and is of type 11', is satisfied; then Jb a

{ L f ( x ) - LSd( f;x ) } ~dx

=

j b{ f(x) a

-

Sd( f; x)} L * L f ( x ) dx.

Theorem 6.10.2. Let f ( x ) in Z Z n ( a ,b) and A : a = x,, < x1 < ... < xN = b be given. Let S,( f ; x) be a generalized heterogeneous spline on d such that S y t ) ( f ;xi) = f("*)(xi)as i ranges over a subset of 0, I , ..., N and ai ranges over a subset of 0, 1, ..., n - 1 which varies with i, and let these interpolation requirements be sufficient to define Sd(f;x). If for any i (i = 0, I, ..., N ) and any a ( a = 0, 1, ..., n - 1) we have S;)(f; xi) # f(c0(xi), let

if i # 0 and i # N , and Pn-u-l(

if i

=

0 or i

=

f

- SA ; xi) = 0

N . Under these conditions,

6.11. Raising the Order of Convergence For most of the generalized splines in which we are interested, it follows from (6.9.1.2), (6.9.3), and (6.9.6) that l f ( - ) ( x )- S(*)(f;x ) ] AN


*

{Kn-1}112 * KnP2 *

***

. Ku

( a = 0, 1,..., TZ - 1).

(6.11.1)

6.11.

207

RAISING THE ORDER OF CONVERGENCE

T h e conditions under which (6.11.1) is valid are considered in Section 6.9, and the constants K and K, ( a = 0, I , ..., n - 1) are defined there. Even if the splines are heterogeneous, (6.1 1.1) applies with 11 d, 11 multiplied by an appropriate constant, provided the irregularities in interpolation and continuity properties are confined to a limited number of meshes in a mesh sequence. If we assume both the validity of (6.11.1) and the second integral relation, it follows by the same argument as that given in Section 5.11 that

- V a b [ f l L . II A , lzn-l-m

( a = 0,

1,..., n

-

1).

(6.11.2)

I n (6.11.2), we have employed the standard notation (6.1 1.3) and the special notation

to illustrate the analogy with Chapter V. From (6.11.2), we obtain Theorems 6.1 1.1 and 6.11.2, which follow.

Theorem 6.11.1. {A,: a

L e t f ( x ) in 3?2n(a, b) and = xoN

= 6) ( N = 1,2,...) < xlN < X mN N

with 11 A , 11 + 0 as N + co be given. If {SdN(f; x)} ( N = 1, 2, ...) is a sequence of generalized splines of dejiciency k such that sAN( f ; x) is a spline on A N with S$j(f;xiN) = f ( a ) ( ~ i (Na) = 0, 1,..., k - 1; i = 1, 2 ,..., mN--l) and, in addition, one of the conditions (a) f o r N = 1, 2, ...,f( x) - SAN(f; X) vanishes at x = a and x = b and is of type 1', (b) f(x) is in Xp2"(a, b), each SAN(f; x) is periodic, andf(a)(xiN)= S$i(f; xiN)( a = 0, I, ..., k - 1; i = 0; N = 1, 2 ,...), (c) for N = I, 2,..., f ( x ) - S,,(f; x) vanishes at x = a and x = b, f ( x ) - SAN(f;x) is of type II', and d l A N , (d) f o r N = 1, 2,..., { f ( a ) ( x )- S d N ( fx)>(-) ; ( a = 0, 1,..., k - 1) and {Lf(x) - LS,,(f; x))(=) ( a = 0, 1, ..., n - k - 1 ) vanish at x = a and x = b, is satisjied, then

c

p y x ) = Sd"'(f; x) N

uniformlj f o r x in [a, 61.

+ O(1l AN jlZn--l--a)

( a = 0, 1, ..., ?t - 1)

VI.

208

GENERALIZED SPLINES

Theorem 6.11.2. Let f ( x ) in X 2 n ( a ,b) and {A,: a

= xON

< xlN < ... < x z ,

(N

= b}

=

1,2,...)

>

withIId,II + O a s N + o o a n d d , C d , ( N l)begiven.Let{SAN(f;x)} ( N = I , 2,...) be a sequence of generalized heterogeneous splines where sAN( f ; x ) is a heterogeneous spline on AN which has deficiency one at each mesh point not in A , and interpolates to f ( x ) at these points. I n addition, for i ranging over a subset of 0, 1,..., m, and ai ranging over a subset of 0, 1, ..., n - 1 which varies with i but is independent of N , let S$(f; x:) = f(m”(xil).If, in the requirements on a heterogeneous spline involving /3r(SA,f; x ) , the latter is replaced by /3,(f - SA,f; x ) , then pyx)

=

S y ( f ;x)

+ O(1l A,

112n-l-m)

( a = 0, 1, 2,...,n - 1)

uniformly for x in [a, b] provided the interpolation requirements are suficient to define each SAN( f ; x).

6.12, Convergence of Higher-Order Derivatives

< x, = b and f ( x ) in X 2 % ( ab) , be given. Let d: a = x,, < x1 < If S A ( f ; x ) is a generalized spline of deficiency k interpolating on d, together with its first k - 1 derivatives, to f ( x ) and its first k - 1 derivatives, then S y ) ( f ;x ) interpolates to f ( = ) ( x at ) least once in every a - 2k 2 mesh intervals if 2k < 01 < 2n - k - 1 and once in every mesh interval for 0 a < 2k. Thus,

+

<

< K , - Ilf‘b+”

-

S~,~’)Ilm . Ij A

11

( a = 0, 1 ,..., 2n - k

- l),

(6.12.1)

where K, is defined as in Section 6.9 but for values of a up to 2n - k - 1 and xai is a suitably chosen point of interpolation of S y ) ( f ;x ) to f(.)(x). Consequently, if we can establish a bound on /I f ( m + l )- St,p’ llco which is independent of the magnitude of 11 d 11, the convergence of S y ) ( f ;x ) to f ( , ) ( x )( a = 0, 1 ,..., 2n - k - 1) as /I d 11 -+ 0 will follow. x x i , we have the representation We know that, for xi-,

< <

(6.12.2)

6.12.

209

CONVERGENCE OF HIGHER-ORDER DERIVATIVES

where the uj(x) and the cij are defined in Section 6.6. Hence, if we can show that the cii are bounded as I/ d 11 + 0, then 11 Ski ]Irn ( a = 0, 1 ,..., 2n - k - 1) is bounded. Since /I f ( m Ijrn ) (a = 0, 1 ,..., 2n - k - 1) is also bounded, the desired convergence will result. T o show the cij are bounded, we revert to the methods of Section 5.12. As in Section 5.12, for some xmiin [xiPl,xi],

amA'

.f

,Xi]

[Xi-l

spy f; Xei).

=

(6.12.3)

Moreover, from Section 6.1 1, we have

ilf

- s A , f I/m

< Bo I

. v ~ b [ f l L //

-

where Bo = K * {Kn-l)1/2 * Kn--2 (6.12.3), we obtain

s-sA ,f [xi-l

c

*.*

zn

,Xi]

=

j-1

(6.12.4)

*

*

K O .As a result of (6.12.2) and

(a = 0 , 1 ,..., 2n -

c i j * u?'(xei)

1; i

=

1, 2,...)N ) .

(6.12.5)

For the left-hand member of (6.12.5), we have the bound

I 8:A,f[xi-l

> xi]i

< {2aRA)b

*

BO

'

Vub"fl,

*

(i = 1 , 2,..., N ;

11 A /12n-m-1 a = 0,

f

1 sfa[Xi-l xill ?

1 ,...,2n

-

l), (6.12.6.1)

where (6.12.6.2) Since max I u r ) ( x i )- U 3ll

1

( ~ ) ( X ~ ~ )

(a = 0, 1,

..., 2n - 1)

approaches 0 as I/ d / / -+ 0 uniformly with respect to both the number and location of the mesh intervals [xi--l, xi] and since the uj(x) are a fundamental set of solutions to L*Lf = 0, it follows that the cij are bounded provided RA is bounded.

Theorem 6.12.1. {A, :a

Let f ( x ) in S 2 % ( ab) , and

= xoN

< xlN < ..* < x z N = b}

(N

=

1,2,...)

with 11 d I\ -+0 as N + co be given. Let {SdN(f; x} ( N = 1 , 2, ...) be a sequence of generalized splines of deficiency k where SdN(f;x) is a spline on A , with SYJf; xi") = f(m)(xiN) ( a = 0, 1 ,..., k - 1 ; i = 1, 2,..., mN - 1) such that one of the conditions, (a) Syi(f;xiN) =f(-)(xiN)( a = 0, 1 , ..., K - 1 ;

210

VI.

GENERALIZED SPLINES

( ~0 ) ( M = 0, 1,..., n - k - 1; 0, m,), {LSA,(f; xiN) - L ~ ( X ~ N ) }= 0, m,) and A , C A , all hold f o r N = 1, 2 ,..., (b)f(x) is in X F ( a , b), S&)(f;xi") = f(w)(xi") ( a = 0, 1,..., k - 1; i = 0; N = I , 2 ,...) and f;x) is periodic, (c) f ( x ) - SA,(f ; x) vanishes at x = a and x = b each SAN( and is of type 1' ( N = 1, 2, ...), (d) f ( x ) - SAN(f;x) vanishes at x = a and x = b and is of type 11' ( N = 1, 2 ,...) and d, C A , ( N 3 l ) , , i s satisjied; then STi(f;x) converges uniformly to f(CO(x) on [a, b] f o r 01 = 0, 1, ..., 2n - k - 1 and

i i

= =

f ( a ) ( x )= S(.)(f;X) AN

+ O(ll A ,

Ilzn- n-k)

provided the RANdefined by (6.12.6.2) are bounded.

Theorem 6.12,2. { A , :a

Let f ( x ) in Z Z n ( a ,b) and

= xoN

< xlN <

< x,"

= b}

(N

=

1,2, ...)

with (1 A,/I -+ 0 as N - + co and A , C d, ( N 3 1) be given. Let { S A N ( fx)}; ( N = 1, 2, ...) be a sequence of generalized heterogeneous splines where SdN(f; x ) is a spline on A , which has deficiency one at each mesh point not in A , and interpolates to f ( x ) at these points. In addition, for cii ranging over a subset of 0, 1 ,..., n - 1 which varies with i but is independent of N , let S y i ) ( f ;xi') = f ( a " ( x t ) If . in the requirements on a heterogeneous N. spline involvzng &(SA,,; x ) the latter is replaced by P,(f - S A , f; x), then we have ( a = 0, 1, ..., 2n - 2) (6.12.7) f ( " ) ( x )= SFd(f;x) + O(I1 A , / 1 2 n - a - 1 ) uniformly for x in [a, b] provided the RAN defined by (6.12.6.2) are bounded and provided the interpolation requirements imposed on each sAN( f ;x) are suficient as dejining conditions.

REMARK 6.12.1. If at the mesh points not in d, we have Syd(f; xi") = 0, 1 , ..., k - 1) and each S A ( f ; x )has deficiency k at these points, then (6.12.7) is valid for M = 0, 1 ,..., 2n - k - 1 but with 2n - 01 - 1 replaced by 2n - 01 - k.

f(a)(xiN)( a =

We can, however, modify the argument used in the proof of the preceding two theorems and obtain the following theorem.

Theorem 6.12.3. or Theorem 6.12.2,

Under the hypotheses of either Theorem 6.12.1

lim Spi(f;x) = f ( " ) ( ~ ) (a = 0, I , ..., 272

N-m

uniformly on [a, b].

-

2)

(6.12.8)

6.13.

LIMITS ON THE ORDER OF CONVERGENCE

21 1

Proof. We know that S;dN,,[xEl , xiN] = Sf=(x2N_1, x i N )

+ O(ll AN

(a = 0,

1Izn-=--l)

1 , ..., 2n - 1). (6.12.9)

I n addition, the boundedness of the cij with respect to N implies that the S y i ( f ; x ) ( a = 0, 1, ..., 2n - 2) are uniformly bounded and equicontinuous on mesh intervals if we take one-sided limits at mesh points where discontinuities occur. Moreover, we can find points xi", and Zi", in any mesh interval [xi--l, xi] such that (6.12.10)

An application of the triangle inequality now establishes (6.12.8), since

+ 8;.

= 8;

6.13. Limits on the Order of Convergence I t may be possible to increase the order of convergence of a sequence of generalized splines ( S d N ( fx)} ; (N = 1, 2 , ...) to a function f ( x ) in CZn[a,b] over that indicated in Theorem 6.12.1, but there is a definite limit. For cubic splines this limit was investigated in Section 3.12 and for polynomial splines of odd degree in Section 5.13. T h e investigation for generalized splines proceeds along similar lines.

Theorem 6.13.1. Let ( A N ) ( N = 1, 2, ...) be a sequence of meshes with 11 AN [I ---t 0 as N - t co and R = supNRAN< 00. Let f ( x ) be in CZn[a,b] and p > 0 . If for each N ( N = 1, 2, ...) SdN(f ; x) is a generalixed spline on A , of dejiciency k and

f(.)

=

)'

+ O(ll

'N

/l""")

(6.13.1)

uniformly for x in [a, b], then L*Lf(x) = 0.

Proof. By the methods of Section 6.12, we can establish under the hypotheses of the theorem that lim S t i ( f ;x) = f ( = ) ( x )

N-tm

( a = 0,

1,..., 2n);

(6.13.2)

VI.

212

GENERALIZED SPLINES

at each step, 2n - 01 - 1 can now be replaced by 2n - 01 consequence, L*Lf(x) = {L*Lf(x) - L*LS,,(f; x)} = 0

+ p. As a

ki

for almost every x. T h e theorem now follows. T h e limitations imposed by Theorem 6.13.1 on an approximated function f ( x ) by an approximating sequence of splines of interpolation are not the only restrictions. I n Chapter IV, we encountered other restrictions of a somewhat similar nature. For generalized splines, they take the following form.

Theorem 6.13.2. Let f ( x ) be in C [ a , b ] , and let {SdN(f; x)) ( N = 1, 2, ...) be a sequence of splines of deJiciency k which interpolate to f ( x ) on a sequence of meshes {A,)(N = 1, 2, ...) with 11 A , 11 -+ 0 as N -+ CO.

If 271

SdN(f; ).

=

1c:qx)

7=1

(xL1

< x < Xi")

and the c c are bounded with respect to N , then f ( x ) is in C2n-k-1[a,b ] .

Proof. It follows from the boundedness of the cg that for 0, 1,..., 2n - k - 1 the sequence {SyJ(f;x)} is uniformly bounded and equicontinuous. Consequently, we can find functions F,(x) such that for a suitable subsequence {A,) ( m = 1, 2, ...) of meshes 01

=

lim S yN), (f;x) =Fe(x)

m-m

(a = 0,

1, ..., 2n

-

K

-

1)

uniformly for x in [ a , b ] . Th u s each F,(x) is continuous. Moreover, since the convergence is uniform, ~ ~ (= x )lim SY) m-m

N,

(j; x) = m-m lim SF) (j; a ) + m-m lim N, = F,(a)

+ lzFa+l(x) dx a

a

~y+l)(fi x) dx N,,,

(a = 0,

1,..., 2n

-

k

- 2).

We conclude that Fo(x) is in CPn-k-l[a, b] and that Fo(x) is identical with f ( x ) , since both Fo(x) and f(x) are continuous and they agree on a dense subset of [ a , b] because of the interpolation properties of the x)}. This completes the proof. sequence {SdN(f; REMARK 6.13.1. T h e requirement that the splines be splines of interpolation to f ( x ) can be replaced by the requirement that they converge to f(x).

6.14.

HILBERT SPACE INTERPRETATION

213

6.14. Hilbert Space Interpretation As we have indicated in Section 5.1, the class X n ( a , b), under the inner product (6.14.1)

is a Hilbert space provided functions differing by a solution ofLf = 0 are identified. Given a sequence of meshes (A,) ( N = 1, 2, ...) on [a, b] with A , C A,,, , we can, as in Chapter V, form the finite dimensional subspaces FAN(k,L ) consisting of all generalized splines of deficiency k on A , and the subspaces [FAN+,(k, L ) - FA,(k, L ) ] consisting of those L ) whose defining values* on A , are zero. Also of inelements in FAN+,(k, L ) - PAN(k,L ) ] terest are the analogous subspaces PAN(k,L ) and [PAN+,(k, of periodic generalized splines as well as the subspaces F;,(k,L) and [Fl,+,(k, L ) - FiJk, L)] of type 11’ splines and the subspaces TAN(k, L) and [TAN,l(k,L) - TAN(k,L)] of type k splines. We can also consider linear subspaces of heterogeneous splines having the same continuity requirements at mesh points but varying defining values. It will be convenient in this case to restrict ourselves to heterogeneous splines that at mesh points not in A , are of deficiency k. We deL ) - FAN(k,L)] note the subspaces analogous to FAN(k,L ) and [FAN+!(k, by HAN(k,L ) and [HAN+l(k, L ) - HAN(k,L ) ] , respectively. I n general, we suppress indication of the dependence on k and L when there is no ambiguity; thus, FAN(k,L ) will be at times denoted by FAN. We emphasize the fact that we obtain different subspaces HA, depending on the continuity requirements imposed on the heterogeneous splines at the mesh points of A , . We now can form the infinite direct sums

* We understand by “defining values,” here and in the remainder of Chapter VI, the prescribable values of the spline and its derivatives.

214

VI.

GENERALIZED SPLINES

T h e orthogonality of the component spaces is established in Section 6.15. 0 as N + co,all these Also in Section 6.15 we establish that, if /I A , /I L ) , are the infinite direct sums, with the possible exception of Pdm(k, same and are identical with X n ( a , b). I n Chapter V, we inferred that, for each integer n > 0, &"(a, b) is dense in X n ( a , b ) under the norm -j

(6.14.3)

Theorem 6.14.1, which follows, establishes that X p n ( a ,b ) is dense in X n ( a , b ) under the norm

+ PnPl(a,b) (6.14.4)

defined by the inner product (6.14.1). Here PnPl(a,b) is the linear space of polynomials on [a, b] of degree n - 1, and X p n ( a , b) + PnPl(a,b) is the linear space spanned by -Xpn(a, b) and Pn-l(a, b).

Theorem 6.14.1. The linear subspace dense in X n ( a , b) under the norm

If L

= Dn, X p n ( a ,b) is dense in

-Xpn(a,b )

+ PnPl(a,b )

is

X " ( a , b).

Proof. Consider the special case where L 3 D", and let f(x) be an arbitrary function in .%?(a, 6). Since D"f(x) is in L2(a,b), Dnf(x) has a Fourier expansion whose partial sums S,(x) are periodic on [a, b]. We can integrate each of these partial sums n times and choose the constants of integration (set them equal to zero) such that the resultant functions F N ( x ) are periodic. Since 11 f - FN jlDn + 0 as N -+ co, the theorem is established for the special case L 3 Dn. We establish the general case by showing that, if n is the order of the differential operator L , then convergence in the norm 11 . ] I D n implies convergence in the norm 11 . ( I L . Since X p n ( a ,b) is dense in X n ( a , 6) under the norm /I \IDn, we can find 7

6.14.

HILBERT SPACE INTERPRETATION

215

a sequence of functions {FN(x)}( N = 1,2, ...) in Xp"(a,b) such that I l f - P , I D n -+ O as N -+ 00. We replace the sequence of functions F N ( x ) by a new sequence {FN(x)) ( N = 1, 2, ...), where FN(x) = F N ( x )+ PN(x).Here PN(x)is a polynomial of degree n - 1 SO chosen that F(m)(a)= f ( * ) ( a () a = 0, 1 , ..., n - 1). With the help of Schwarz's inequality, we are led to the following system of inequalities:

where the constants Ki( j = 0,..., n - 1) are independent of N . From the definition of L [ f ( x ) - F N ( x ) ]and Minkowski's inequality, together with inequalities (6.14.5), we know that

This implies that I/ f - FN llL -+ 0 as N + co and establishes the theorem. It has been asserted in Chapter I11 that Y 2 ( a ,b) is a Hilbert space under the norm 11 * llD2 , in Chapter V that %"(a, b) is a Hilbert space under the norm / / * / I D n , and in the present section that X n ( u , b) is a Hilbert space under the norm / / * (IL. Once allowances are made for the pseudo character of the norm, all properties but the completeness of the spaces under the respective norms are immediate. This latter property is also easy to establish.

Theorem 6.14.2.

II ' IIL

The space Y n ( u , b) is complete under the norm

*

Proof. Let (f,} ( N = 1, 2, ...) be a Cauchy sequence in %"(a, b) with respect to the norm 11 [IL. Then {LfN}( N = 1,2, ...) is a Cauchy 4

216

VI.

GENERALIZED SPLINES

sequence in L2(a,b). Since L2 is complete, this sequence has a limit g(x) in L2(a, b). If f(x) is any solution of Lf(x) = g(x),* then f(x) is in X n ( a , b) and // f N - f ItL ---f 0 as N + 00; thus, %?a(, b) is complete under the norm 11 /IL .

-

6.15. Convergence in Norm Theorem 6.15.1. Let f(x) in X n ( a , b ) and {A, :a

= XoN

< X N < *.. < X mN N

=

b} ( N

=

1,2,...)

with A N C A,,, and / / d, I/ + 0 as N + CO. Let (SdN(f; x)} ( N = 1, 2, ...) be a sequence of generalized splines of deficiency k which interpolate to f(x) on d and, in addition, f ( a ) ( x i N )= S$J(f;x i N ) ( a = 1, 2, ..., k - 1; i = 1 , 2,..., mN - 1; N = 1 , 2,...). Ifone ofthe conditions, (a) f ( x ) - S d N ( fx) ; is of type I' ( N = I, 2,...), (b) f(x) - S A N ( fx) ; is of type 11' ( N = 1, 2,...), (c) f ( a ) ( x , N ) = St)(f;xiN) ( a = 1, 2,..., k - 1 ; i = 0, m N ;N = 1, 2,...) and each S,(f; x) is of type k , (d)f(a)(xiN)= S$i(f;xiN) ( a = 1,2,..., k - 1; i = 0 ; N = 1, 2, ...), f(x) is in -Xpn(a, b), and each SdN(f; x) is periodic, is satisfied, then lim Ilf - s+,f II = 0. N+w

x) can be regarded as a spline of interpolation Proof. Since SAN(f; to S d N + M (x)f ;( M 3 l), the sequence {I1 SdN(f; x) /IL} ( N = 1, 2,...) is monotonic increasing and is bounded above by /If / I L . This is a consequence of the first integral relation, which also implies that

I t follows that the sequence {SdN(f; x)) is a Cauchy sequence with respect to the norm / j . /IL , since the sequence {I/ SAN,r /IL } converges. Moreover, by the same argument as used to establish (6.9.6), we have

I/ S A N + M . ~- S d ~ , f lID" d K l l

S d ~ + ~ . f S d N . f lIL

for (1 A N 11 sufficiently small and some positive constant K that is index)) is a Cauchy sequence in pendent of N . Consequently, (Si:,",'(f; L2(a,b ) . If g(x) is the limit in L2(a,b ) of this sequence and G(x) = f ( n - l ) ( a )

+ jzs(x) dx, a

* T h e solution can be taken in the sense of Caratheodory (cf. Sansone and Conti [1964, p. 111).

6.15.

217

CONVERGENCE I N NORM

then j SY;l)(j; x)

-

G(x)l

< (6 - a ) 1 / 2 a {Syd(j;x) g(x)}2dx/ + j SF;"(f; a) - f ( n - l ) (a)I,

1/2

-

(6.15.1)

and, as N -+ co, the right-hand member of (6.15.1) approaches zero. It follows that f ( " - l ) ( x ) is identical with G(x) on [a, b ] ; hence,

In addition, we have the inequalities

+

n

( b - ~ ) ~ + ' - *j I Sp;j)(f; a) -f(+j)(u)l, (6.15.3.n)

i=l

which, in turn, imply the inequality (for N sufficiently large)

where K is a positive constant independent of N . T h e theorem now follows. T h e same argument can be applied to a sequence of heterogeneous splines; Theorem 6.15.2 is the result.

Theorem 6.15.2. {A,:

a = XoN

Supposef(x) is in X " ( a , b) and

< X I N < ... < x,;

= 6)

(N

=

1,2,...)

VI.

218

GENERALIZED SPLINES

with A , C A,,, and 11 A , 11 + 0 as N -+ 00. Let { S A N ( fx)} ; ( N = 1, 2, ...) be a sequence of generalized heterogeneous splines satisfying the hypotheses of Theorem 6.9.2. Then

klzllf

- SANJ

IIL

= 0.

Lemma 5.15.1 and Theorem 5.15.3 have direct analogs for generalized splines; in particular, we have the following lemma.

Lemma 6.15.1. Let A , and A , be two meshes on [a, b] with A , C A , , and let Sdl(x) and S,,(x) be two generalized splines on A , and A , , respectively, each having the same quantities prescribed on A , and the same continuity requirements on A , . If the prescribed values (defining values) of SA2(x)at the mesh points of A , are zero, then (Sdl,

Proof. (Sdl

SdJL =

Let A , be defined by a

, S&

= U

=

xo

0.

< x, <

< X,

=

b. Then

LSd2(x). LSdl(x) dx

the final equality holding because, whenever a term in

is not cancelled directly by another term, because of continuity, it is multiplied by a defining value of SA2(x)on A , and is thus zero. T h is proves the lemma, which in turn establishes Theorem 6.15.3.

Theorem 6.15.3. [a, b] with A , C A,,,

Let {A,} ( N = 1, 2, ...) be a sequence of meshes on and such that / I A , I\ + 0 as N + CO. Then the

6.16.

CANONICAL MESH BASES

219

in$nite direct sums (6.14.2) are orthogonal decompositions with respect to (f,g)L and, with the possible exception of PAm(k, L ) , are identical with < X n ( a ,b ) . In the case of PAm( k , L ) , b ) c PA&

%"(a,

L ) c P y a , b).

6.16. Canonical Mesh Bases T h e definition and construction of canonical mesh bases for FAm(k, L), FL m ( k , L ) ,TAm ( k , L ) , H A m ( k , L ) , and PAm ( k , L ) is identical with the construction already given in Chapter V and will be omitted here. Both the statement and proof of the analog of Lemma 5.16.1, however, require some modification. T h e proof is more complicated to the extent that we need to show that, if S,(L; x) is an element of a canonical mesh basis associated with the linear differential operator L of order n, then

II S&

x)Ilo*


*

/I Si(G 4 l l L

(6.16.1)

for some positive constant K that is independent of i ; the analog of the inequalities (5.16.2) will involve this constant. We proceed as in Section 6.9. Because of the vanishing of the defining values of Si(L;x) at the mesh points of preceding meshes in the mesh sequence {Ti)( i = 0, 1,...) arising in the construction of the canonical mesh basis, we can take f(x) as the zero function; thus, we modify (6.9.1) by setting

J z = J b (S',")(L;x)j2 dx.

(6.16.2)

a

By essentially the same method of argument as in Section 6.9, however, the existence of a positive constant K such that (6.16.1) holds is clear. Consequently, there exist positive constants K, independent of i such that the analog of Lemma 5.16.1 is valid. Since these constants admit no simple representation, we do not state them explicitly.

Lemma 6.16.1, Let {S,(L;x ) } ( i = 1, 2, ...) be a canonical mesh basis f o r [FA ( k , L ) -FAl(k,L)], [FLm(k,L ) - F i l ( k , L ) ] , [TAm(k,L) - TA1(k,L)], [H,_(k;L) - ffAl(k, L ) ] , or [PAm(k, L ) - P,l(k, L ) ] determined by a sequence of meshes {A,) ( N = 1, 2 ,...) with A , C A,,, . L e t {nil (i = 0, 1,...) be the related sequence of meshes used in the construction of {Si(L;x)). Then there exist positice constants K, , independent of i , such that

I Sy)(L;x)/

< K, /I

lj(2n-2u-1)/2

(a = 0, I ,..., n - 1).

(6.16.3)

VI.

220

GENERALIZED SPLINES

From the preceding lemma, we obtain the following analog of Theorem 5.16.2.

Theorem 6.16.1. Let {S,(L;x)} (i = 1, 2, ...) be a canonical mesh basis such that 11 ri11 = O( l/i). Then there exist real numbers pa > 0 such that m

S,(X) =

1 {Sp'(L;x ) } ~< /3=

(a = 0,

1,...,n

-

2).

(6.16.4)

i=l

T h e elements S,(L; x) of a canonical mesh basis not only have the property of orthogonality, but if d, C A,,, the basis for is simply an extension of the basis for FA,. If d, is defined by a = x,, < x1 < . - * < X, = b and if, to obtain a basis for FA,, we employ cardinal splines Si(d, ; x) whose defining values at all but the mesh point xi vanish and there only one defining value is nonzero, we not only lose orthogonality, but in passing from FAN to a completely new set of basis elements is needed. There is a definite analogy here between this situation and the use of Newtonian interpolation formulas rather than Lagrangian interpolation formulas. When additional interpolation points are added, a whole new set of Lagrangian unit functions is needed (Davis [1963, p. 411); on the other hand, the set of Newtonian functions can be supplemented to accomodate the new interpolation points.

6.17. Kernels and Integral Representations I n this section, we state the analogs for the case of generalized splines of the theorems contained in Section 5.17. T h e statement of the theorems, however, is such that heterogeneous splines are included. T h e proofs differ in no essential way from the earlier proofs and are consequently omitted.

Theorem 6.17.1. Let f(x) be in %&(a, b ) or Xpn(a, b), and let {Si(L,k ; x ) ) (i = 1, 2 ,...), together with {ui(L, k ; x)} (i = 1, 2 ,..., m), be a canonical mesh basis for Fdm(k,L),F i m ( kL, ) , Tdm(k, L ) , HAm(k,L ) , or PAm(k, L ) , which is an orthonormal basis for X " ( a , b) or Xpn(a,b). Then for every x in [a, b] H,,,(L, k; x, t ) Lf(t)dt

+ G(,)(x)

( a = 0, 1,

...,n - 1)

(6.17.1)

6.18.

APPROXIMATION OF LINEAR FUNCTIONALS

22 1

whereLG(x) = 0 for x in [a, b], Ha*,&, k; x, t )

c u y ( L , k; m

=

x)

*

Luz(L, k; t )

i=l

+ c S y L , k; x) N

LS,(L, k; t ) ,

(6.17.2)

i=l

and the limit exists uniformly with respect to x in [a, b].

Corollary 6.17.1. Let f ( x ) be in X 2 n ( a ,b) or Xp2"(a,b), and let {S,(L, k ; x)} (i = 1 , 2 ,...), together with {ui(L,k ; x ) } (i = 1, 2 ,..., m), be a canonical mesh basis (cf.footnote Section5.17) for FAw(k,L ) ,FAm(k,L), , is an orthonormal basis for T A m ( k , L ) ,H A m ( k , L ) ,or P A m ( k , L ) which X R ( a ,b) or .Xpn(a,b). Then for every x in [a, b] and a = 0, 1 , ..., 2n - 2, Eq. (6.1 7.1) is valid. I n the case of heterogeneous splines, the maximum deficiency is assumed not to exceed k . Theorem 6.17.2. Let (n-,} (i = 0, 1 , ...) be a sequence of meshes on [a, b] determining a canonical mesh basis forFAm(k,L ) ,Fim(k,L), TAm(k, L), Hdm(k,L),or PAm(k,L),which is an orthonormal basis for X n ( a , b) or Xpn(a,b).If H,,N(L, k;x, t ) is defined by (6.17.1.2), then for each X { H , , ~ ) (a = 0, 1 , ..., n - 2; N = 1 , 2 ,...) is a Cauchy sequence in L2(a,b) and, consequently, a Cauchy sequence in L(a, b). If H,(L, k ; X , t ) ( a = 0, l,,.., n - 2 ) denotes the common limit, then for f ( x ) in *%(a, b) or Xpn(a, b) (6.17.3) f ( , ) ( x ) = H J L , k; x, t ) * Lf(t)dt G("'(x), a

+

where L G ( x ) = 0 for x in [a, b]. Moreover, the convergence is uniform with respect to x in [a, b], and Ha(& k ; x, t ) is obtainedfrom H,-,(L, k ; x , t ) by formal term-by-term differentiation with respect to x.

REMARK6.17.1. We are tacitly assuming, when we have a canonical mesh basis for X n ( a , b) or Xpn(a,b), that 11 T, 11 = O ( l / i ) . 6.18. Representation and Approximation of Linear Functionals Analogs of the four theorems contained in Section 5.18 remain valid for generalized splines, and again the arguments needed to prove the theorems for generalized splines are essentially unchanged. Con-

222

VI.

GENERALIZED SPLINES

sequently, we again content ourselves with just the statement of the theorems and omit the proofs. We do, however, consider some examples of approximating linear functionals in which we approximate an integral using these equally spaced values of the integrand. These same examples are considered by Sard [1963, Chapter 111, and they illustrate the manner in which spline theory provides many of the "best approximations" obtained by Sard. We also connect generalized spline theory to the calculation of the eigenvalues of a linear differential operator. We conclude this section with an application of heterogeneous splines to the approximation of point functionals LZZ:f -+f(x).

Theorem 6.18.1. Let f ( x ) be in X a ( u , b) or -Xpn(a,b), and let {S,(k,L ; x)} (i = 1, 2 ,...), together with {u,(k,L ; x)} (i = 1, 2 ,..., m), be u canonical mesh basis f o r FAm(k, L ) , Fim(k,L ) , TAm(k, L), HAm(k, L ), or P,,(k,L), which is an orthonormal basis for -Xn(a, 6 ) or Xpa(a,b). If 9 is a lineur functional of the f o r m (.5.18.1), then HN(k,L ; t ) Lf(t)dt

+ Y o G,

(6.18.1 . l )

where

1 u?'(k,L; b

ai = 3-0

pi =

a

c j S?'(k,L; 'I

b

j-0

a

s) dpj(s)

(i = 0, 1 ,..., m), (6.18.1.3)

s) dpj(s)

(i = 1, 2,...,N ) , (6.18.1.4)

and LG(x) = 0 for every x in [a, b].

Corollary 6.18.1. Let f ( x ) be in X Z n ( a b) , or Xp(a,b), and let {S,(k,L ; x)} (i = 1, 2,...), together with {u,(k,L;x ) } (i = 1, 2,..., m), be a canonical mesh basis for FAm(k, L), Fim(k, L), TAm(k, L), HAm(k, L), or PAm(k,L), which is an orthonormal basis f o r %"(a, b) or ,Xp"(a, b). If 9is a linear functional of the f o r m 9 0

f

=

5 r f " ' ( t ) d&),

j=o

(6.18.2)

a

where each p j ( t ) is a function of bounded variation on [a, b], then (5.18.1) holds except that in this case 71 < 2n - k - 1.

6.18.

223

APPROXIMATION OF LINEAR FUNCTIONALS

REMARK6.18.1. I n both the theorem and the corollary, the function

G(x) is dependent on the function f ( x ) , but the kernels H,(k,L; z) are

not. T h e method of proof essentially depends on the uniform convergence of the spline sequence and its derivatives to f ( x ) and its derivatives, and not on the rate of convergence. Thus, in view of Theorem 6.12.3, we need only require 7 < 2n - 2 rather than 2n - k - 1 in Corollary

6.18.1.

Theorem 6.18.2. Let f ( x ) be in %"(a, b) or .&"(a, b), and let {S,(k,L ; x)} (i = 1, 2 ,...), together with {ui(k,L ; x)} (i = 1, 2,..., m), be a canonical mesh basis f o r FAm(K, L), Fim(k,L ) , TAm(K, L), HAm(k,L), or PAm(k, L ) , which is an orthonormal basis f o r X n ( a , b) or X p n ( a ,b). I f 9 is a linear functional of the f o r m (5.18.1) except that q n - 2, then {HN(k,L ; x ) } is a Cauchy sequence in L2(a,b) and, consequently, a Cauchy sequence in L(a, b). I f H(k, L ; x) denotes the common limit, then

<

9of =

/ H ( k , L ;t ) *Lf(t)dt + Y o G, b

a

(6.1 8.3)

where LG(x) = 0 f o r every x in [a, b]. Turning from the representation of linear functionals to the approximation of linear functionals, we have the analogs of Theorems 5.18.3 and 5.18.4.

Theorem 6.18.3. I f 2 ' is a linear functional of the f o r m (5.18.1) and if B is the linear functional such that B Of

=3 0

(6.18.4)

SA,~,

S d ( f ;x) being the spline of interpolation to f ( x ) which is a linear combination of &(k, L ; x)} (i = 1, 2,..., m) and G(x), then N

* L f ( t )dt,

(6.18.5.1)

where

[ f ( x ) is in Y n ( a , b) or Xpn(a, b)], and the pj(s) ( j = 0,1, ..., q) are functions of bounded variation on [a, b] determined by 9 alone through the representation (6.18.1).

VI.

224

GENERALIZED SPLINES

Theorem 6.18.4. If 9is a linear functional of the form (5.18.1) and B is a linear functional of the form (6.18.4) with 7 < n - 1, then H ( k ,L ; t ) * Lf(t)dt.

H(k, L ; t ) is in L2(a,b) and is the limit, in the mean square sense, of

where

86

c1 SY’(m r l b

=

j=1

a

s)

44)

(7 < n);

(6.18.6.2)

the pi(s) are functions of bounded variation on [a, b] and depend on 9 alone [(Againf ( x ) is in Y n ( a , b) or Xpn(a,b).)] If B is any linear functional of the form (5.18.9) and 9- B annihilates solutions of L f ( x ) = 0, then for k < n Zof-Bof

=

[

b

J a

H(K,L;t).Lf(t)dt,

(6.18.7)

where H(k, L ; t ) is in L2(a,b) and is defined as in Theorem 6.18.2 with 9 replaced by 9 - B. Moreover, the integral (6.18.8)

is minimized if B is of the form (6.18.4)

REMARK6.18.2. If the coefficients aij defining B in (5.18.9) are subject to additional constraints, it may not be possible to satisfy (6.18.4). Generally, however, this can be done if heterogeneous splines are used. This is particularly true if certain of the aij are required to be zero. , or XF(a,b), the restriction 7 < n Furthermore, if f ( x ) is in Y Z n ( a b) imposed by (5.18.1) can be replaced by the condition 7 < 2n - 2 in Theorem 6.18.3. We remind the reader that the deficiency k of a spline is never allowed to exceed n, the order of the operator L. We now consider the approximation of an integral of the form 9 0f

=

s’

-1

by functionals of the form

f ( x ) dx

(6.18.9)

6.18.

225

APPROXIMATION OF LINEAR FUNCTIONALS

We could equally well consider an arbitrary interval

[a, b]

and take (6.18.11)

but the calculations would become more cumbersome. Moreover, there is no loss of generality in the simplification (6.18.9), since

so that

which is of the form (6.18.11). Let us impose the additional requirement that the approximation is exact for linear functions. Then by either Theorem 5.18.4 or Theorem 6.18.4. (6.18.12) and (6.18.13)

is minimized when B o f = Y o SA,,, where cubic spline of interpolation to f ( x ) on A : - 1 ul(x)

then

=

1,

uz(x) = x,

= x72,

sA(f;

x) is the type 11’

< 0 < 1. Let

u4(x) = 9 / 6 ; (6.18.14)

VI.

226

GENERALIZED SPLINES

But we have

1 1

0

f 0

uz(x)dx = 8

1 1’ 1 0

0

so

ul(x) dx = 1,

u3(x) dx = 6

ul(x) dx = 1,

-1

,

-1

,

-1

s”

u4(x) dx = & ,

uz(x) dx =

- 4,

u3(x) dx = Q , u4(x) dx =

-1

-

,

so that

The interpolation properties of SA(f;x) imply that f ( 0 ) = c11 = c21 f(-l) = c11 - c12 f(1) = cz1 c22

+ + + + 9

&l, - QC1a

kc23

9

Qcu, 9

whereas from the continuity of SA’(f;x) and Si(f;x) at x c12 = cZ2 and cI3 = cz3 . Finally, since the spline is of type 11’, we must have 0

=~

13 ~14,

0

+

=z ~ 2 3

As a result, C13

= CB = C14 =

and 3 0 SA,f

from which it follows that

= 2c11

+

-C=

kc13

.

~ 2 4

=

0 we have

6.18.

APPROXIMATION OF LINEAR FUNCTIONALS

227

and which in turn implies (6.18.16)

This is the rule of 3 - 10 - 3 obtained by Sard [1963, p. 421 through direct minimization of (6.18.13). If we require B to be exact for quadratic functions of x, we obtain a better known approximation with less effort. In this instance, 9 0f

- B Of =

J”’

-1

H&)

Pf(t) dt,

(6.18.17)

and (6.18.13) is minimized when B of = 2’0 S A , f ,S A ( f ;x) being the type 11’ quintic spline of interpolation to f ( x ) on d . Since

interpolates to f ( x ) on d and is a type 11’ quintic spline on d, the uniqueness of SA(f;x) asserts that y(x).

sA(f;

Consequently,

which is simply Simpson’s rule. Again the result was obtained by Sard through direct minimization of (6.18.13). In addition, Sard showed that, if B is exact for cubics, the remainder has a representation 2 o f - B 0f

=

I

1

H&) D”f(t)dt

(6.18.19)

-1

for which (6.18.13) is still minimized when B is given by Simpson’s rule. From the standpoint of spline theory, this gives rise to a serious difficulty, since the type 11‘ polynomial spline of degree seven is not uniquely defined on A . The quadratic y(x) defined by (6.18.18) is one possibility, and 2’0y does yield Simpson’s rule. If, however, we add an arbitrary fourth point p to d to form a new mesh 6, then the type 11’ polynomial spline of degree seven SA(f; x) is uniquely defined and is the

228

VI.

GENERALIZED SPLINES

cubic polynomial of interpolation to f ( x ) on 6. I n addition, if = 90 Sd,f.,B is exact for cubics, (6.18.19) is valid, and this choice of B minimizes (6.18.13). Let

B of

U(X)

=

x3-x.

~

P3-P ’

then u ( x ) is a cubic polynomial that vanishes on A and has the value 1 atp. Moreover, S d ( h 4 = Y ( 4 M P ) -r(P)>U(X)? and 90 Sd,f = 2’0 y (f(p) - y(p)} 90 u = 90 y. Consequently, we have verified the well-known result that Simpson’s rule is exact for cubics, and we are still in agreement with Sard that, given an approximating functional of the form (6.18.10) which is exact for cubics, the remainder can be represented in the form (6.18.19), and (6.18.13) is minimized when B is identical with Simpson’s rule. I n addition, we have shown that, if we add an arbitrary fourth point p to A , distinct from the mesh points of A , and consider approximating functionals B of the form 3 Of = .!(-I) bf(0) cf(l) aP),

+

+

+

+

+

we cannot improve the approximation with respect to the measure (6.18.13). T h e previous application required only polynomial splines; the next application involves generalized splines. Consider the problem of determining the eigenvalues of a linear differential operator L of the form (6.1.1), subject to a set of n linearly independent auxiliary conditions, each specifying thatf(x) or one of its first n - 1 derivatives vanish at some point in [a, 61. We restrict ourselves to a simple case in which the auxiliary conditions involve only the endpoints of the interval. Thus, we require f‘4(a)=0 (i = 1, 2,..., I; aj # aj , i # j ) , (6.18.20) fW(b) =0 ( j = 1,2,***,J ; yi # yj , i # j ) ,

+

and I J = n. I n addition, the ai and yi are less than n. Let A : a = xo < x 1 < ... < x N = b be given, and let .% denote the family of simple generalized splines S,(L; x ) associated with the operator L, which satisfy the boundary conditions (6.18.20) together with the additional conditions pn-,,-&Sd

1,2)...,I),

;u ) = 0

(i

;b ) = 0

(i = 1,2,..., I ) ,

=

(6.18.21)

6.18.

229

APPROXIMATION OF LINEAR FUNCTIONALS

=0 and y satisfies (6.18.20), then y = 0. Before considering the eigenvalue problem determined by L and the constraints (6.18.20), let us consider the problem of finding a solution to the nonhomogeneous equation

&(v; x) being defined by (6.3.2). Moreover, we assume that if Ly

Lf(X)

=, ).(g

a

< x < 6,

(6.18.22)

which satisfies the constraints

+

where we have I J = n and the ai and y j are less than n. If the homogeneous equation has a solution satisfying (6.18.23), the nonhomogeneous problem is solved if we can find a particular solution of (6.18.22) that satisfies the homogeneous boundary conditions (6.18.20). We observe that TAis an inner product space under the inner product

Let the dimension of FA be m, and let {.,(I,; x)) (i = 1, 2, ..., m) be an orthonormal basis. Let {Pi}(i = 1, 2, ...) be a sequence of distinct points in [a, b] which are distinct from the mesh points of A. We require that the sequence {Pi}be such that, if we form a sequence of meshes {TJ (i = 0, 1,...) by adding the points Pidne at a time in their enumerated order starting with no = A, then 11 ri11 = O(l/z]. We can consequently introduce additional orthonormal splines S,(L; x) which satisfy the boundary conditions (6.18.20) and (6.18.21), with the result that {S,(L;x)} (i = 1, 2,...), together with {ui(L;x)} (i = I, 2,..., m),comprise a canonical mesh basis for X R ( u ,b). Let f ( x ) satisfy (6.18.22) and (6.18.20). Then, by Theorem 6.17.2, if the boundary conditions (6.18.20) uniquely determine a solution of the differential equation (6.18.22), we have f(X)

=

J

b

a

H(L; x, t)Lf(t) dt;

(6.18.24)

here H(L; x, t) is the mean square limit of {HN(L;x, t)} ( N where m

H,(L;

X,

t ) = C ui(L;X) * Lui(L;t ) i-1

N

+ C Si(L; X) i=l

*

=

1, 2, ...),

LS,(L; t ) . (6.18.25)

230

VI.

GENERALIZED SPLINES

Moreover, from Lemma 6.16.1 we know

< K(l/iY-1’2

I S,(L; ).I

(i = 1, 2, ...).

(6.18.26)

T h e solutionf(x) of (6.18.22) and (6.18.23) is now given by f(X) = G(x)

+ 1 H(L; b

a

x, t ) g ( t ) dt,

(6.18.27)

where G(x) is a solution of the homogeneous problem. We can interpret the kernel H{L;x, t ) as a Green’s function. T h e eigenvalue problem (6.18.28)

Lo(x) = X V ( X ) ,

where ~ ( x )satisfies the conditions (6.18.20), similarly is equivalent to the eigenvalue problem .(X)

=

xJ

b

a

H ( L ; x, t ) v ( t ) dt.

(6.18.29)

Here,we have the important advantage that H(L; x, t ) is the mean square limit of the degenerate kernels H,(L; x, t ) provided n 2 2. I n this case, an approximate solution can be obtained by solving a problem where the kernel is degenerate. T h e inequality (6.18.26) is useful in justifying the approximation and estimating the rate of convergence. The preceding application of generalized splines is not limited to the rather special boundary conditions defined by (6.18.20). The main requirement on the boundary conditions is that they be such that the first integral relation is valid. Given a mesh A : a = x, < x1 < < xN = b and a linear differential operator L, we require for the validity of the first integral relation that N i=l

P[f-

sA,f

, L s ~ , f ]

Ixi

xi-1

= 0,

(6.18.30)

where P[u, v ] is the bilinear concomitant of L. Observe now that the continuity and interpolation requirements imposed on SA(f;x) at the interior mesh points of d reduce (6.18.30) to P [ f - s A , f ,L s A , f ] l :

= O.

(6.18.31)

Consequently, if f ( x ) - S A ( f ;x) satisfies a linear homogeneous set of n boundary conditions sufficient to imply that (6.18.22) has a unique

6.18.

23 1

APPROXIMATION OF LINEAR FUNCTIONALS

solution on [a, b], and if LS,(f; x) satisfies an adjoint set of boundary conditions, then (6.18.31) holds, and the theory goes through as before. Another important observation is that, if we are given a self-adjoint differential operator R = L*L, where L contains no singularities on [u,b], then we can apply the same methods to solution of the equation W(x)

(6.18.32)

= g(4,

subject to a set of self-adjoint linear homogeneous boundary conditions, provided the boundary conditions imply the validity of (6.18.31). This will be the case, for instance, if we require f y u ) = f ( = ) ( b )= 0

(a = 0,

1,..., n - 1).

(6.18.33)

We then have

; t)} ( N in this case, H ( R ; x, t) is the uniform limit Of { H N ( R x, where

=

1, 2,...),

The proof of (6.18.34) is analogous to that of (6.18.24). We need only observe that in this case Jb

a

LS,(L; x ) L f ( x )dx

=

r

a

S,(L; x)L*Lf(x) dx.

Similar remarks apply to the associated eigenvalue problem

M x ) = hf(x). When R is of the second order, L is of the first order; consequently, the inequality (6.18.26) is too weak to establish the uniform convergence of H N ( R ;x, t) to H ( R ;x, t). However, if in obtaining our canonical mesh basis we add new mesh points in sweeps moving from left to right, at each step of the sweep the new point bisecting a mesh interval, uniform convergence can still be established. The important fact here is that the basis elements S,(L; x) vanish identically except in the two adjacent mesh intervals separated by the mesh point in the associated mesh ri at which S,(L; x) does not interpolate to zero. This follows from the uniqueness of the spline S,(L; x) and the fact that, since L is of first order,

232

VI.

GENERALIZED SPLINES

only Si(L;x) but not its derivatives is required to be continuous on [u, b]. Thus, if S,(L; x) does not vanish identically on any mesh interval at both ends of which it vanishes, then uniqueness is contradicted, since a second spline with the same interpolation and continuity properties as S,(L; x) results when the definition of S,(L;x) is altered so that it vanishes on the mesh interval in question. Thus, given x in [a, b], there are only two basis elements in each sweep which do not vanish at x. Since in each sweep all mesh intervals are halved, uniform convergence follows from Lemma 6.16.1. We conclude this chapter with an example in which the approximation of a linear functional 2 requires heterogeneous splines. The example comes from numerical analysis, more specifically from the numerical integration of ordinary differential equations. Assume that our integration steps are of equal length. Since the dependence on step size can be shown to be linear, we assume unit length. Consider two consecutive intervals defined by three points x,-~, x, , x,+~ which we can take as 0, 1,2, respectively. Thus, given the mesh 0 < 1 < 2, we desire a corrector formula for estimating the value of a function f ( x ) at x = 2 from its values at x = 0, x = 1 and the values of its first derivative at x = 0, x = 1, x = 2. Moreover, we desire the approximation to be exact for cubics. [Here we are assuming that we have obtained satisfactory values of f ( x ) and f ’ ( x ) at x = 0 and x = 1, and, with the aid of a predictor formula and the differential equation, we have obtained an acceptable value of f ’ ( x ) at x = 2 for use in the corrector formula.] Finally, we desire that the corrector formula be a linear function of the five quantities f(O), f(I), f’(O), f’( I), f’(2) and that these quantities reflect the total dependence of the formula on the function f ( x ) . Our general theory now tells us that we can approximate the linear functional dpo f = f(2), subject to the preceding conditions, if we employ a heterogeneous polynomial spline S,(f;x) of degree seven. We then have

where the remainder 9 ( 2 ) can be represented as w(2) =

f N(2, t ) Pf@) dt. 2

(6.18.36)

0

Moreover, the integral (6.18.37)

6.18.

233

APPROXIMATION OF LINEAR FUNCTIONALS

is minimized by taking Sd(f; 2) as the approximating functional. T h e following set of conditions completely defines Sd(f;x): S,(f;O) =f(O),

S;(f;0 ) = f ’ ( O ) ,

S,(f; 1) =f(l);

q f1);=f’(l),

qf; 2) =f’(2);

( a = 0, 1, 2, 3, 4, 5 ) ; sy(f;1+) = sy(f;1 -) sy(f;0 ) = 0 ( a = 4, 5 ) ;

S$’(f; 2) = 0

( a = 4,

i

(6.18.38)

5, 7).

If ui(x) (i = 1, 2, ..., 8) are the pol’ynomials xi, and we let SA(f;

).

= cI1ul(x)

SA(f;

).

= c21u1(x)

+ +

‘1Zu2(’) E22u2(X)

+ + + + ”’

c18u8(x)

**.

c2&(x)

< < 1, 1 < < 2,

0

%’

then the conditions (6.18.38) yield a set of 16 independent linear equations for obtaining the 16 quantities cij in terms of f(O), f( l), f’(O), f’(l), and f’(2). Their exact solution results in the predictor formula

obtained by Sard [1963, p. 831, and, as he points out, it is stable and convergent in the sense of Dahlquist [1956]. Numerical solution of the system of equations arising from the conditions (6.18.38) gives the approximate coefficients

-

-

ii

__

-

0.1860464 0.8139536 0.3488372

/

1.209302 0.2558140.

I

(6.18.40)

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CHAPTER VII

The Doubly Cubic Spline

7.1. Introduction Our development of spline theory has been one-dimensional u p to this point; the theory, however, generalizes readily to higher dimensions. Just as the simple cubic spline is of fundamental importance in onedimensional spline theory, the simple doubly cubic spline (Section 1.2) is basic in two-dimensional spline theory. We limit ourselves to the consideration of simple two-dimensional splines in this chapter; a very general theory is developed in Chapter .VIII. Suppose that we are given a rectangular region 9: a t b; c s d of the plane. Then, if we are given two one-dimensional meshes A , : a = to < t, < < t, = b and A , : c = so < s1 < < sM = d, the resulting two-dimensional mesh rr = {Pij}(i= 0, 1,..., N ;j = 0, 1,..., M ) , where Pij = (ti , si), partitions 9 into a family of subrectangles

< <

< <

= 1 , 2 ,..., M ) .

A simple doubly cubic spline S,(t, s) on 9 with respect to rr is (1) a double cubic in each rectangle g i j ,and (2) an element of CZ4(9),where CTn(R) is the family of functionsf(t, s) on 9 whose nth order partial derivatives, involving no more than rth order differentiation with respect to a single variable, exist and are continuous. As in the one-dimensional case, we can distinguish between a spline and its representations. It is possible to express a two-dimensional spline as a linear function of a finite set of linearly independent parameters; the choice of these parameters, or defining values as we often refer to them, is far from unique. I n this chapter, we confine ourselves to a limited but important group of these representations. A type1 representation of a spline includes among the defining values of the spline the following values of its partial derivatives: 235

236

VII.

THE DOUBLY CUBIC SPLINE

(a)

at at the mesh points {Pij}

(i = 0 , N ; j

(b)

asat the mesh points {Pij}

(i = 0 , 1,..., N ; j

PS,

(c)

at the mesh points {Pij]

( i = 0, N ; j

= 0,

1,..., M ) , = 0, M ) ,

= 0, M ) .

T h e remaining parameters needed to represent S,(t, s) uniquely at times may not be specified, but, for splines of interpolation on T , they are the values of S,(t, s) at the mesh points of T . I n a similar fashion, we define a type 11 representation as one including among the defining values of a spline the values of (a)

?S, at2 at the mesh points {Pij}

(b)

at the mesh points { P i j }

(i=O,N; j = O , l ,

(i = 0, 1,..., N ; j

..., M ) ,

= 0, M ) ,

A two-dimensional spline is periodic in t if S,(t, s), aS,(t, s)/at, and PS,(t, s)/at2 are periodic functions of t with period b - a. T h e definition of a spline periodic in s is analogous. A doubly periodic spline is periodic in both t and s with period b - a in t and d - c in s. T h e most convenient set of defining values for a doubly periodic spline consists of its values at the mesh points {Pij>(i = 1, 2 ,..., N ; j = 1 , 2,..., M ) , and this is what is implied by the terminology a doubly periodic spline of interpolation on T . At times, the words doubly or on ir may be suppressed. If the values of partial derivatives specified by (7.1.1) are zero, we speak of the spline as a typeI‘ spline, and if the values of partial derivatives specified by (7.1.2) are zero, we speak of the spline as a type 11’ spline. As in the one-dimensional case, it is desirable torgo a step further and separate functions f ( t , s) defined on 92 into type I and type 11 equivalence classes. Th u s two functions are in the same type I equivalence class if the partial derivatives specified by (7.1.1) are defined and equal. A function for which the quantities in (7.1.1) are defined and zero is a type I’function. T h e equivalent definitions for typcII and type II’functions are immediate. We point out, however, that, unlike the one-dimensional analogues, these definitions are mesh-dependent, since they depend on both the number and spacing of the mesh points on the boundary of 9. Historically, two-dimensional spline theory lagged its one-dimensional

7.2.

PARTIAL SPLINES

237

counterpart by more than 15 years. Birkhoff and Garabedian [I9601 attempted a generalization, but it represented only a beginning. A year later, DeBoor [I9621 inaugurated two-dimensional spline theory by establishing the existence of the simple type I spline of interpolation to a function f ( t , s) on a rectangular mesh. T h e minimum norm property, best approximation property, convergence properties, and orthogonality properties were obtained by Ahlberg, Nilson, and Walsh [abs. 1964a; 1965b] in 1964. Before developing the theory of doubly cubic splines further, we consider a useful generalization.

7.2. Partial Splines

<

Let a rectangle 9!:a < t b; c < s < d , together with a mesh A , : a = to < t , < < t , = 6 , be given. I n addition, we assume that we are given N + 1 functionsfi(s) ( i = 0, I , ..., N ) defined on the interval c s d . For each s, we can form the one-dimensional type 11’ cubic spline SAt(Y(s);t ) , where

< <

y(s)= [fO(S),

fi(s),*..,fN(s)lT’

(7.2.1)

I n the same manner if f o ( s ) = f,(s), we can form the periodic spline S,t(Y(s); t ) , or, if we are given two additional functions go(s) and g,(s), we can form type I and type I1 splines, where the additional derivatives with respect to t at t = a and t = b, for a given value of s, are specified by the functions g,(s) and gN(s),respectively. For each s, we have the minimum norm property and the best approximation property holding in a one-dimensional sense. If we require the functionsfi(s) ( i = 0, I , ..., N ) , go(s),and g,(s) to be in L(c, d ) , then, since SAt(Y(s);t ) depends linearly on these functions and since they reflect its total dependence on s, it Y(s);t ) / a t and a2SAt(Y(s); t ) / a t 2 are all in follows that SAt(Y(s);t ) , asAt( L(c, d ) . Under these conditions, Theorem 7.2.1, which follows, is typical of the type of theorem valid for partial splines; its proof is immediate from one-dimensional spline theory. Aside from a few remarks regarding convergence and the effect of linear operators on partial splines, we leave the translation of one-dimensional spline theory into a theory for partial splines to the reader.

Theorem 7.2.1. Let N + 1 functions f i ( s ) (i = 0 , I,..., N ) , each in L(c, d ) , be given, together with a mesh A , : a = to < t , < < t , = b. Then of all functions f ( t , s) defined on the rectangle W : a < t < b;

238

VII.

THE DOUBLY CUBIC SPLINE

< <

c s d , which coincide with fi(s) at t = ti ( i = 0, 1 , ..., N ) and which are in X 2 ( a ,b) for each s in [c, d ] , the type 11' partial spline SAt(Y(s);t ) , where Y(s)is given by (7.2.1), minimizes

and is the unique admissible function that minimizes this integral. As we refine the mesh A , , for each s the functions SAt(Y(s); t ) and s) and af(t, $ ) / a t ,respectively, uniformly with respect to t . Furthermore, if

asdt( Y(s);t ) / a t converge to f(t,

is uniformly bounded as a function of s, the convergence will be uniform with respect to s as well. I t is of importance to note that, because of the linear dependence of S,t(Y(s);t ) on the functionsfi(s) (i = 0, 1,..., N ) , go(s), and gAr(s), we have a ~ s , ~ ( ~ (t)jass);

=

s,,(a-u(s)/a~; t)

for all a: for which iYY(s)/as. is defined. Indeed, this applies in general to linear operators, provided they are well defined. As a further example, we have

s" C

SAt(Y(s);t ) ds

= SA*

(jY(S)ds; t ) . d C

I n what follows, we speak of simple partial splines. We understand by this terminology that the induced one-dimensional splines obtained by fixing the variable s are simple splines.

7.3. Relation of Partial Splines to Doubly Cubic Splines T h e concept of a partial spline allows a very direct approach to the construction of doubly cubic splines which reduces the construction to the construction of one-dimensional splines. We proceed as follows. Let 9: a t b; c s d be given, and let f ( t , s) be defined on 9. If a mesh x on 9 is defined by A , : a = to < t , < < t , = b and A , : c = so < s1 < < sM = d and if the partial derivatives occurring in (7.1.1) exist for f ( t , s), then we can set

< <

< <

- 7 .

f&)

=f(ti

, s)

(i = 0, 1 ,...,N ) , (7.3.1)

7.3.

RELATION OF PARTIAL SPLINES TO DOUBLY CUBIC SPLINES

239

and, for i = 0, 1,..., N , construct the type I spline S,Jfi ; s) of interpolation to fi(s) on d, such that f i ( s ) - SAJf i ; s) is of type 1’. I n addition, let SdJgi ; s) (i = 0, N ) be similarly defined for go(s)and gN(s), respectively. Now SAJ f i ; s) (i = 0, 1,..., N ) , together with SAjgi ; s) (i = 0, N ) , defines a type I partial spline, denoted S,(f; t, s), which is easily verified to be a simple doubly cubic spline of interpolation to f(t, s) on T. One need only observe that, for tiP1 t ti ,

< <

k=O

where Aij(t)( j = 0, 1, ..., N ) and Bij(t)( j = 0, N ) are cubic functions s sj ( j = 0, 1,..., M ) , we know that S,,Jfi ; s) of t. But for si-l (i = 0, 1, ..., N ) , SAJgi ; s) (i = 0, N ) are cubic functions of s; cont ti ; sjP1 s si sequently, in each rectangle gij: ti-l (i = 1, 2,..., N ; j = 1, 2,..., M ) , S,( f;t , s) is a double cubic. I t is in C2*(9)by the nature of its construction. T h e preceding construction has established the existence of at least one simple type I doubly cubic spline of interpolation to f(t, s) on T such that f(t, s) - - S,( f;t , s) is of type 1’. This, however, immediately raises the question of uniqueness, since a second such spline can be obtained by interchanging the roles of s and t. I n order to establish uniqueness, we first extend the minimum norm property to doubly cubic splines; once this is done, a simple uniqueness argument can be given. Uniqueness can also be established through a more careful examination of Eqs. (7.3.2). T h e construction of periodic and type I1 splines of interpolation to f(t, s) on 9? proceeds in the same manner. We observe that the cubics in s or t, to which S,(f; t, s) reduces on the boundary of each rectangle gij,* furnish sufficient information to determine S,( f;t , s) on gii ; these cubics can be obtained from the one-dimensional splines used in the construction of S , ( f ; t , s). Since the values of f(t, s) and certain of its partial derivatives are required only at a finite number of points, f(t, S) is essentially an arbitrary function on 9.

< <

< <

< <

* This statement needs to be qualified to the extent that quantities such as a2S,/as at at the vertices of 9 c are j obtained by constructing the one-dimensional splines of interpolation to a s / & along the grid lines s = st-* and s = s1 and then differentiating with respect to t.

240

VII.

THE DOUBLY CUBIC SPLINE

7.4. The Fundamental Identity

< < < <

We assume thatf(t, s) is in I?:(%) and that the rectangle W :u t 6; is partitioned into subrectangles Wii : tiPl t ti ; Sj-l<S<Si (;= 1,2 ,..., N ; j = 1,2 ,..., M ; t O = U , t N = b , S o = C , sM = d) by a mesh T . We have the obvious identity c

<s
(7.4.1)

We modify the last term in the right-hand member of (7.4.1) by integrating it twice by parts with respect to t. We then have

and, after two more integrations by parts this time with respect to s,

7.4.

THE FUNDAMENTAL IDENTITY

24 1

If we substitute this result into (7.4.1), we are led to the identity

(7.4.2) I n this identity, which we call the fundamental identity, S,(t, s) is a double cubic in each rectangle gii( i = 1 , 2,..., N ; j = 1, 2,..., M ) , but we have not assumed that S,(t, s) is in CZ4(92).If the latter assumption is made, S,(t, s) is then a simple doubly cubic spline on 9, and (7.4.2) reduces to

(7.4.3) I t is important to note that, although we cannot expect a5Ss(t,s)/as2 at3 to be continuous at a grid line t = t i , both its left-hand and righthand limits as t -+ ti are continuous functions of s. A similar observation

242

VII.

THE DOUBLY CUBIC SPLIN E

applies to a5S,(t, s)/as3 at2 with the roles of s and t interchanged. T h is fact has been used in the reduction of the fundamental identity to the form (7.4.3), which it assumes for simple doubly cubic splines.

7.5. The First Integral Relation Under a variety of auxiliary conditions, the fundamental identity reduces to the integral relation

(7.5.1)

which we call the first integral relation. Again, this is a direct generalization of the one-dimentional situation. For simple doubly cubic splines of interpolation, we have the following theorem.

< <

< <

Theorem 7.5.1. Let g:a t b; c s d be given along with a mesh n- defined by A , : a = to < t, < < t, = b and A, : c = so < s, < < sM = d. Let f ( t , s) be in C',4(L%?). If S,( f ; t, s) is a simple doubly cubic spline of interpolation to f ( t , s ) on rr such that one of the conditions, (a) f ( t , s ) - S,( f ; t, s) is of type 1', ( b ) S,( f ; t, s ) is of type IT, (c) f ( t , s) and S,( f ; t , s ) are doubly periodic, is satisfied, then

---

Of course, the first integral relation is valid under other end conditions. 7.6. The Minimum Norm Property As a direct corollary of Theorem 7.5.1, we obtain a generalization to doubly cubic splines of the minimum norm property of one-dimensional splines. We formalize this in Theorem 7.6.1, which follows.

< <

< <

Theorem 7.6.1. Let 9:a t 6; c s d be given along with a mesh n- defined by A , : a = to < t , < * * . < t , = b and A, :

7.7.

UNIQUENESS AND EXISTENCE

243

< sM = d. In addition, let { f i j } (i = 0, 1, 2,..., N; c = so < s1 j = 0, 1 , ..., M ) be a prescribed set oj' real numbers, where fii is associated with the mesh point ( t i , sj). Then of all functions f ( t , s) in CZ4(a)the type II' spline of interpolation to the values fii on rr minimizes (7.6.1)

and is the unique admissible function that minimizes this integral. I f , in addition, af(t, s)/& is prescribed at (ti , sj) (i = 0, N ; j = 0, 1,..., M), af(t, s)/i3s is prescribed at ( t i , sj) (i = 0, 1 ,..., N; j = 0, M ) , and 8 f ( t , s)/& 3s is prescribed at (ti , sj) (i = 0, N ; s = 0, M ) , then (7.6.1) is minimized by the corresponding type I spline of interpolation. I f foi = f N j ( j = 0, 1,..., M) and fio = fiM (i = 0, 1 ,..., N ) , then (7.6.1) is minimized by the doubly periodic spline of interpolation. In both these cases we also have uniqueness. As indicated, the fact that (7.6.1) is minimized follows from the first integral relation. T h e latter also implies that any other function g ( t , s) minimizing (7.6.1) differs from the spline of interpolation by a linear function of s and t ; it follows from the interpolation requirements that the linear function vanishes identically.

7.7. Uniqueness and Existence With the establishment of Theorem 7.6.1, we have provided an easy means of demonstrating the uniqueness of type I, type 11, and doubly periodic, doubly cubic splines of interpolation, for the argument can now proceed along the same lines as in the one-dimensional case.* T h e doubly periodic spline of interpolation is typical. Thus, let S,(f; t, s) and f ; t , s) be two doubly periodic splines interpolating on a mesh T to a function f ( t , s) defined on a rectangle 9.Then their difference S,( f;t , s) f;t, s) is a doubly cubic spline that interpolates to the zero function Z ( t , s) on T ,and consequently it follows from the minimum norm property that S,( f ; t, s ) - S,(f; t, s) must differ from Z ( x ) by a solution of

s,(

s,(

ayp2 at2

= 0,

that is, by a linear function of s and t. Interpolation requirements then ensure that qf; 4 s) = S , ( f ; t , 4.

* For splines possessing the minimum norm property, uniqueness is an immediate corollary of Theorem 7.6.1.

244

VII.

THE DOUBLY CUBIC SPLINE

<

Theorem 7.7.1. Let 9:a t \< 6 ; c \< s < d be given along with a mesh T defined by A , : a = to < t, < < tN = b and A, : c = so < s1 < < sM = d. If f ( t , s) is defined on 9 and ;f S,( f; t , s) is a simple doubly cubic spline of interpolation to f ( t , s) on T such that one of the conditions, (a) f ( t , s) - S,( f;t, s) is of t y p e I ’ , ( b ) f ( t ,s) - S,( f;t, s) is of type 11‘, (c) f ( t , s) and S,( f;t , s) are doubly periodic, is satis-ed, then S,( f;t , s) is unique.

REMARK 7.7.1. Since f ( t , s) is only specified at mesh points, Theorem 7.7.1 asserts that in any type I or type I1 equivalence class there is at most one spline of interpolation to f ( t , s) on T . T h e existence proof in Chapter 111 for one-dimensional cubic splines consists of exhibiting a linear system of equations which ensures that the interpolation and continuity requirements imposed on S,( f;t , s) are satisfied, and of then using uniqueness to show that the system of linear equations is solvable. I n two-dimensions, it is not trivial to write down in a straightforward manner a finite system of linear equations which will ensure the desired continuity of S,( f;t , s) and its derivatives along grid lines. Existence, however, follows from the construction of S,( f;t , s) contained in Section 7.3, since the latter requires only the existence of one-dimensional splines. Once existence has been established, the doubly cubic nature of S,( f;t , s) in each mesh rectangle gij allows an appropriate system of independent linear equations (cf. Ahlberg et al. [1965b]) to be formulated. 7.8. Best Approximation Because of the linear dependence of a spline on its defining values, we have, under appropriately chosen end conditions, the decomposition S,( f + g; t , s)

= S,(

f;t , $1 + S&

t,

(7.8.1)

is satisfied. If we apply the decomposition (7.8.1) to f ( t , s) - S,(t, s),

7.9.

245

CARDINAL SPLINES

where S,(t, s) is an arbitrary simple spline with respect to virtue of the minimum norm property we have

J:

Jl

{ f ” ( t , S) =

-

j:1:

T,

then in

Y ( t ,s ) } ~dt ds

{ f ” ( t , s) - S”(t,s)

-

S:(f

-

Sn; t , s)}‘ dt ds

+ s” J’” {S”(f - Sn; t , s ) } ~dt ds e

=

j:j b

a

{ f ” ( t , s) - S:(f; t , s))’

dt ds

+ !”: j b( S ” ( f a

-

S,, ; t , s ) } ~dt ds,

provided end conditions are such that S,(S, ; t , s) = S,(t, s). This argument establishes the best approximation property of simple doubly cubic splines.

< <

< <

Theorem 7.8.1. Let a rectangle 9: a t b; c s d be given along with a mesh T defined by A , : a = to < t, < < t , = b and A, : c = sfl < s, < < sM = d. Then i f f ( t , s) is in C24(9),S,(t, s) is any simple spline with respect to ir, and S,( f ; t, s) is the simple spline of interpolation tof ( t , s ) on T such that f ( t , s ) - S,(f; t , s) is of type 1‘,we have

1:I” a

( f ” ( t , S) - S;(t, s)}’

dt ds

3

11

Jb

a

{f”(t,

S)

- S ; ( f ; t , s)}’ dt ds,

(7.8.2)

and S,( f;t, s) is unique in this sense up to a solution of a”f(t,s ) / a s 2 at2 = 0. If all functions are required to be doubly periodic or to be in a prescribed type II equivalence class, then (7.8.2) is valid i f S,( f;t , s ) is interpreted as the corresponding spline of interpolation. In the doubly periodic case, S,( f;t , s ) is unique up to a constant; otherwise, uniqueness is up to a solution of a y ( t , $)/as2 a t 2 = 0.

7.9. Cardinal Splines Cardinal splines have occurred elsewhere in this book (cf. Section 2.7); in interpreting the representation (7.3.2) for a doubly cubic spline, they again serve a useful purpose. I n a very broad sense, we can define a cardinal spline to be a spline for which exactly one defining value is one and all others are zero. T h e functions A j ( t )= Aij(t), B,(t) = B,(t), BN(t)= BiN(t),

< t < ti < t < ti tiPl < t < ti

(i = 1, 2,...)N ; j = 0, 1,..., N ) , (i = 1, 2,...)N ) , (i = 1, 2)...)N ) ,

1

(7.9.1)

VII.

246

THE DOUBLY CUBIC S P L I N E

which enter into (7.3.2), are one-dimensional cardinal splines. We have, however, the one-dimensional representations

I

( i = o , l , ..., N ) ,

c M

SAs(g0; s)

=

at

j=O

+ Bo(s)

a”f(to

as at

sM)

9

as at

9

1

(7.9.2)

where the functions &s), &(s), and BM(s)are the corresponding one-dimensional cardinal splines defined by the mesh d, : c = SO

< 51 <

* * a

< SM = d.

If the representations (7.9.2) are used in (7.3.2), we obtain

+

F~M(t,

as at’

+ FNM(t,

S)

’

as at

where CZj(t,s) = Ai(t)Aj(S)

D i j ( t ,s)

= Bi(t)Aj(S)

E i j ( t ,s)

= Ai(t)B j ( s )

Fii(t, s)

= Bi(t)&(s)

(i = 0, I , ..*,N ; j = 0, 1)...,M ) , (i = 0 , N ; j = 0 , 1,..., M ) , (i = 0, 1,..., N ; j = 0 , M ) , (i = 0 , N ; j = 0,M ) .

, (7.9.3.1)

I

(7.9.3.2)

7.10.

CONVERGENCE PROPERTIES

247

T h e functions Cij(t,s), Dij(t, s), Edi(t,s), Fij(t,s ) are clearly the twodimensional cardinal splines needed to represent s,( f ; t , s), and as (7.9.3.2)demonstrates, they are obtainable by forming pairwise products of one-dimensional cardinal splines. Thus, although S,( f ; t , s) is not normally representable as the product of a function of t and a function of s, it is always representable as a finite sum of such products.

7.10. Convergence Properties T h e convergence properties of doubly cubic splines are virtually immediate consequences of the representation (7.3.2) which may be written as

with the aid of the notation of (7.9.1). Consequently,

and lim

II A t /I -0

lim

II A , II -0

S,(f; t , s) = f ( t ,

s).

Iff(t, s) is in C,4(a),convergence is uniform with respect to both t and s, and the iterated limit can be taken as the double limit. By differentiating S,( f ; t, s ) and using similar arguments, we obtain the following theorem.

< <

< <

Theorem 7,lO.l. Let 92: a t 6; c s d be given, < t, = b and together with a mesh T deJined by A , ; a = to < t , < A, : c = so < s1 < *.. < sM = d. If f ( t , s ) is in C24(9?),and 11 n-11 -+ 0, then a Y & ( f ;t , s)/asa atb converges to aYf(t, s)/as. atb uniform& with respect t o t ands f o r y = a +/3 ,< 2, a 1, /3 1 , and

<

I f f ( t , s) is in C48(92),then

<

248

VII.

THE DOUBLY CUBIC SPLINE

+ <

<

<

moreover, (7.10.3) is validfor y = OL /3 6 , OL 3, /3 3, provided the ratios of the mesh norms to the minimum distance between adjacent mesh points are uniformly bounded as 11 A , 11 and I( A , 11 approach zero; indeed, under these conditions the exponents 3 - fl and 3 - 01 may be replaced by 4 - /3 and 4 - a , respectively.

REMARK7.10.1. T h e end conditions required of S,( f;t , s) are essentially dictated by one-dimensional spline theory. I n particular, Theorem 7.10.1 is valid if S,( f;t , s ) is a spline of interpolation to f ( t , s) on rr such that f ( t , s) - S,( f;t, s ) is of either type I' or type 11'. If both f ( t , s) and S,( f;t , s ) are doubly periodic, the theorem is valid. For (7.10.2) to apply, the splines S,( f;t , s) need only be restricted to a fixed type I1 equivalence class. 7.1 1. The Second Integral Relation There is an analog of the second integral relation for one-dimensional splines which applies to doubly cubic splines. It asserts that under suitable end conditions

however, since convergence properties can be established directly from the analogous convergence properties of one-dimensional splines via the representation (7.10. l), its importance is diminished. Theorem 7.1 1.1 gives several conditions under which (7.1 1.1) is valid; its proof proceeds along the same general lines as the proof used to establish the first integral relation for doubly cubic splines.

< <

< <

Theorem 7.11.1. Let a rectangle 9: a t b ; c s d be given along with a mesh rr defined by A , : a = to < t, < *.. < t, = b and A, : c = so < s1 < < sM = d. Let f(t, s ) be in C4*(9)and S,,(f; t, s) be a spline of interpolation to f(t, s) on T . If one of the conditions, (a) f ( t , s) - S,( f ; t, s) is of type I', (b) f(t, s) - S,(f; t , s ) is of type II', ( c )f(t, s) and S,( f;t, s ) are doubly periodic, is satisfied, then - a -

7.12.

THE DIRECT PRODUCT OF HILBERT SPACES

249

7.12. The Direct Product of Hilbert Spaces Let H , and H z be separable Hilbert spaces with bases {ei} (i = 1, 2, ...) and {fj}( j = 1 , 2, ...), respectively, and let denote the linear space generated by the quantities {h,) = {eifj> (i = 1, 2,...; j = 1,2,...). If we define (hij , hkdB = (ei ek)Rl - (fj ,fi)R2 (7.12.1) 9

?

are the inner products associated with H I and (., where (., and H , , then (., can be extended by linearity to an inner product for R. Let (7.12.2) I1 u llR = (u, + Y 2 ; then we can complete R with respect to /I . (la and extend the inner product (., .)a to the completion of f?. We denote the latter by H and - ) H . It is readily verified that H is the extended inner product by a separable Hilbert space with respect to (., . ) H . A more detailed discussion of this construction is given by Sard [1963, p. 3541. Our motivation for discussing the direct product of Hilbert spaces is to form the direct product of X 2 ( a ,b) and X 2 ( c , d). I n Chapter 111, we considered X 2 ( a , b) as a pseudo-Hilbert space under the pseudoinner product (a,

(f,g) = / h ) g " ( t )

dt,

and we determined several orthonormal bases for X 2 ( a ,b). T h e most important of these bases for our purposes are the canonical mesh bases. Let {ui(t)}(i = 1, 2, ...) be a canonical mesh basis for X 2 ( a ,b) and {'uj(s)) ( j = 1, 2, ...) be a canonical mesh basis for X 2 ( c , d). If @: a t b; c s d denotes the rectangle determined by (a, b) and (c, d), we let X2(93) denote the direct product of X 2 ( a ,b ) and Y 2 ( c , d). It follows that

< <

< <

(7.12.3) is the proper inner product for X z ( 9 )since ,

=

1' a

u:(t) u;(t) dt

1'

$'(s)

vi(s) ds.*

C

We now show that CZP(92) C X2(9).

* This relation justifies interpreting .P(.% as a)space of classes of functions on 9 and the symbolic product u,vl as the pointwise product of the functions u,(t) and vj(s). We must, however, identify functions differing by a linear function o f t and s.

250

VII.

THE DOUBLY CUBIC SPLINE

Let { r N(}N = 1, 2, ...) be a sequence of meshes on 9 defined by AtN: < t,N < < tEN = b and A , N : c = soN < sIN < < s2N = d, where rTTM refines rNif M > N , and let {S,(f; t , s)} ( N = 1, 2, ...) be an associated sequence of splines of interpolation on these meshes to a functionf(t, s) in C24(9).Since, for M > N , SN(f;t, s) is a spline of interpolation to S,( f;t, s) on r N, it follows, if the first integral relation applies, that a = to"

Therefore, the sequence of real numbers

is monotonically increasing and is bounded above by

consequently, the sequence

is a Cauchy sequence in L2(%) and has a limit g(t, s) in L 2 ( 9 ) Proceeding . as in Chapter 111, we let

We then have

We can now conclude from Theorem 7.10.1, the definition of g(t, s), and Schwarz's inequality

7.13.

THE METHOD OF CARDINAL SPLINES

25 1

from which it follows that

Consequently, we have

< <

< <

t 6; c s d be given, together Theorem 7.12.1, Let 9: a with a sequence of meshes (rN}( N = 1 , 2,...), where rr, is dejined by d f N : a = t,N < tlN < .-.< tzN = b and AsN: c = soN < slN < < sgN = d for each N. Let )I rr, 11 -+ 0 as N + co and rNrefine r N f for N' < N.* If f ( t , s) is in C24(9)and ;f one of the conditions, (a) f ( t , s) - S N ( f ;t, s) is of type I' ( N = 1,2,...), (b) each S,( f;t, s) is of type I.' ( N = 1, 2,...), (c) f ( t , s) and each S,( f;t, s) are doubly periodic, where SN(f;t, s) is a spline of interpolation to f(t, s) on 7rN , is satisjied, then lim

N+w

i l f - sN,f

/ / X 2 ( R )=

O'

Corollary 7.12.1, C24(9)C X 2 ( R ) . 7.13. The Method of Cardinal Splines I n Section 7.9, doubly cubic cardinal splines were constructed by multiplying together pairs of cubic cardinal splines; in this section, we exploit this and other properties of cardinal splines in order to approximate, under suitable conditions, solutions of second-order partial differential equations. Let 9: a t 6; c s d and f ( t , s) in C 4 * ( 9 )be given. If {ui(t,s)} (i = 1, 2,...) is an orthonormal basis for .X2(9) which is obtained by forming the direct product of canonical mesh bases for X 2 ( a ,b) and X 2 ( c , d ) , respectively, then

< <

If 2 W

-

b a

iIX2(*)

< <

a"f(t, s) a4u4t, s) dt ds

a s a~t 2

as2 at2

* We can define 11 rNI) as max{ll A t N [ ) I/, A s N ) ( } .

(7.13.1.1)

= O,

(i = 1, 2,...).

(7.13.1.2)

VII.

252

THE DOUBLY CUBIC SPLINE

Moreover, if the constants b, , b, , b, , b, are properly chosen, we have

+ b,t + 6,s + b4ts + c m

f ( 4 S)

= 6,

(7.13.2)

W i ( t , S).

i=l

If we let N

S N ( f ;t , S)

= 6,

+ 6,t + 6,s + b4ts + C

api(t,S)

( N = 1, 2,...)

(7.13.3)

i=l

and then restrict ourselves to a suitable subsequence, the SN(f ; t , s) are splines of interpolation to f(t, s) to which Theorem 7.10.1 applies. Thus, for y = 01 p, where 0 01, /3 6 4 andf(t, S) = SN(f;t , S) E N ( f ;t , s), we have

+

+

<

In Eq. (7.13.4), A t N and AsN are intended to denote one-dimensional meshes arising in the construction of the canonical mesh bases from which u,(t, s) (i = 1, 2, ..., N ) were obtained. We observe that, if we assumef(t, s) to be in C2(B) rather than in C4*(B),then the proof of Theorem 7.10.1 holds in view of Theorem 2.3.4, and we have convergence for 0 01, /3 < 2; but the rate of convergence is slower, and we cannot estimate the error in as precise a fashion. Thus, although we use (7.13.4) in demonstrating convergence, our results apply under much broader circumstances. Assume that we are given a partial differential equation of the form

<

4 ut,

+

a&,

S)

+

$1 us, + adt, 4 us, + a4(t, S) ut

us + %(t, 4 u + g(t, 4 = 0

(7.13.5)

defined on 9, together with a set of boundary conditions such that the problem is well set in the sense of Hadamard. Thus, we assume that (7.13.5) has a unique solution U ( t ,s) which depends continuously on the prescribed boundary data. Since we are basically interested in demonstrating a method, we assume that the partial differential equation is elliptic and that U(t,s) is prescribed on the boundary of 92 by a function f ( ~ )T, being the arc length along the boundary, such that f ’ ’ ( ~is) continuous along each side of 9. Hencef(7) also defines U,(t, s) along vertical sides of 93’and U,(t,s) along horizontal sides of W . Finally, we assume that the coefficients in (7.13.5) and boundary data are

7.13.

253

THE METHOD OF CARDINAL SPLINES

sufficiently smooth so that U(t, s) is in C z ( 9 ) . T h e assumption of “ellipticity” and “smoothness” does not obscure the generality of the method and allows a clearer exposition. If for N = 1, 2, 3, ... we let SN(U ; t, s) be the spline of interpolation to U(t,s) such that U ( t ,s) - SN(U ; t, s) is of type I’ for each N , then in the notation of Section 7.9 we can write SN(U ; t , s) as

cc

“N “ N

S,( U ; t , s)

=

2=0

j=o

Uti

, Sj) a t ) &(s)

+ C {Ut(a, mN

sj)

j=O

+ 1 (U,(ti + ‘) “N

?

t=O

d,

Ust(‘r

+ Ut(b,

sj)

BnN(t))Aj(s)

4 Bo(s) + US(ti 4 Bm”)

A&)

!

BO(t)

$-

Bo(t)

’Ofs)

$-

BON(t) BWLN(s)

+

1‘

B?lN(t) B O ( s )

dl B % N ( t ) ‘mN(’)’ (7.13.6)

We can rewrite (7.13.5) in the form LU

+ g(t, s ) = 0

(7.13.7.1)

if we define a2

L = a,(t, s) - . at2

+

a,(t, s)

a

+ a,(t, s) __ + a,(t, s) - . as at

-

at

a2

a2

a

.

+ a,(4

a

7 & *

as2

+ ad4 s)

*.

(7.13.7.2)

t , s),

(7.13.8)

Consequently, LSN(U;

t , S)

=

-g(t,

S) - L E N ( U ;

and the term LEN(U ; t , s) can be made arbitrarily small in magnitude in view of (7.13.4) by choosing N sufficiently large. If we evaluate the members of (7.13.8) at the mesh points (ti , sj) (i = 0, l,..., n N ; J’ = 0, 1,..., mN), we obtain (nN+ l)(mN 1) linear equations for the unknowns

+

u(ti , sj)

Ut(ti ,sj) U s ( t i ,sj) U,qt(ti 9

sj)

(i == 1 , 2, ..., n~ - 1 ; j = 1, 2,***1m N (i = 0, nN ; j = I , 2 ,..., n t N - I), (i = 1, 2 ,..., nN - 1; j = 0, m,), (i = 0, njq ; j = 0, m N ) .

-

I), (7.13.9)

254

VII.

THE DOUBLY CUBIC SPLINE

T h e remaining quantities U(ti , si), U , ( f i si), , U,(ti , si) entering into (7.13.6) are determined by the boundary data. I n matrix form, we have the equations ANU,

=G N

+

E N ,

(7.13.10)

where A, is the resulting matrix, U, is the vector of unknowns, GN is the vector arising from -g(t, s) and the boundary data, and EN is the vector arising from -LE( U ; t, s). Although the existence of is not guaranteed, it is likely, since U ( t ,s) exists and is unique, and since S,(U; t, s) is uniquely determined by U ( t ,s). If we assume A;l to exist and 11 AslEN 11 + 0 as N -+ co, then the solutions of the approximate equations

uN

ANDN= G ,

(7.13.11)

determine doubly cubic splines SN(t,s) which converge to U (t ,s) as N - + 00. T h e rate of convergence is determined by (7.13.4) and by the behavior of the matrix Ail. Observe that the S,( U ; t , s) do not satisfy the boundary conditions exactly, but only spline approximations of the boundary data; thus, it is important that the problem be “well set.” For situations where 9 is partitioned into approximately 100 or less subrectangles, the matrices A, can be inverted numerically by elimination methods; otherwise, iterative techniques are normally required. Since the Ai(t),Aj(s), Bi(t),and Bj(s) are cubic splines, their determination is greatly simplified. T h e resulting approximations are not completely numerical, since Eq. (7.13.6), or a more convenient rearrangement, provides intermediary values and can be used in analytic investigations. This latter feature makes the method attractive for ordinary differential equations as well. Since the error is given by A ; l E N , error estimates can be calculated. When a considerable number of intermediate points are needed for a precision plot of the output, the method of cardinal splines is advantageous over methods of equal accuracy which yield only the values of U(t,s) at mesh points.

7.14. Irregular Regions U p to this point, we have confined ourselves to doubly cubic splines defined on rectangular regions. Our results extend readily, however, to other regions. One of the simplest extensions is to regions that are the union of a finite number of rectangles. I n defining type I or type I1 representations for such regions, the same quantities are specified along the sides of the rectangles parallel to the t axis as are specified by (7.1.1)

7.14.

IRREGULAR REGIONS

255

or (7.1.2), respectively. T h e same is true with respect to the sides of the rectangles parallel to the s axis. T h e quantities specified by (c) of (7.1.1) or by (c) of (7.1.2) must be prescribed at each corner of the rectangles. T h e first integral relation, the minimum norm property, the best approximation property, as well as all the other properties obtained for doubly cubic splines on a rectangular region, can be immediately generalized to regions of this type. We leave this development to the reader and turn to the consideration of regions, or perhaps more appropriately “surfaces,” which under suitable mappings are mapped onto a rectangular region B. T h e mappings are not 1 - 1 (in fact they are not single-valued mappings), and so not all functions on 9 correspond to functions on the original surface. Let S denote the surface depicted in Fig. 1. We assume that we are of smooth curves in S , given a point Po in S together with a family {TQ)

FIG. 1

each connecting Poto a point Q of I‘,the boundary of S. We assume that I’ is also a smooth curve in the sense that each of its describing coordinates has a continuous second derivative with respect to arc length, that the correspondence between the boundary points Q and the curves rQ is 1 - 1, and that any pair of these curves has only the point Po in common. T h e family of curves {TQ) is “surface-filling” in the sense that, except for P o , there is precisely one curve of our family passing through any point P of S. Let d(P, P’) denote the distance between P and P’ measured along the curve rQ passing through P. T h e distance function d(P, PI) is clearly not defined for an arbitrary pair of points P and P‘ of S, but, if P and P’ can be joined by a curve I’, , this can be done in only one way so that d(P, P’) is well defined in this event. Also, we let t(P)denote the distance measured along I’ in a counterclockwise sense from a fixed point Qo of r to Q p , where Q p is the unique point of I’ such that P lies on r,, . Now let s(P) be defined by (7.14.1)

256

VII.

THE DOUBLY CUBIC SPLINE

Thus, to each point of S not on rQo, we can assign a unique pair of we have two numbers [t(P),s(P)]. With each point P # Po of rQ0 distinct pairs: [0, s(P)] and [Z, s(P)],where 1 denotes the length of r. For P o , we have the infinity of pairs ( t ,0), where 0 \< t 1. Thus, the mapping F : P [t(P),s(P)],although it is not single-valued, maps S onto the rectangle 9: 0 t I; 0 s 1. Under the mapping F, the curve T Q Q , # Qo , is mapped onto the line segment t = t(Q) parallel to the s axis. T h e curve rQo is mapped onto both the line segments t = 0 and t = I, the point Po is mapped onto the line segment s = 0, and the curve is mapped onto the line segment s = 1. Observe that, although the mapping F is. not single-valued, it has a single-valued inverse mapping F-l, since each point ( t , s) in 9 is the image of precisely one point of S. Consequently, with each function f ( P ) defined on S we have the unique image function f(t, s) = f [ F - l ( t , s)]. Because of , certain partial derivathe smoothness of the curve r and the curves rQ tives o f f ( P ) are expressible in terms of the partial derivatives off(t, s), and conversely. Thus, with a function f ( P ) defined on S , we can associate splines of interpolation S( f;.P) which are in reality splines of interpolation to the corresponding functions f(t, s) = f [ F - l ( t , s)] defined on 9.Clearly, it is the functions f(t, s) in CZ4(9)which are constant on s = 0 and periodic with respect to t that are important in this context. A variation of the mapping just described is also very useful. Suppose that, in place of the family of curves rQjoining Po to the boundary points Q , we have a surface-filling family of smooth curves {TQ,Q,}, where both Q and Q f are on We consider two cases: ( 1 ) through each point P of S except for two points Q,, and Q1 on r there is a unique curve I'o,o, of our family; and (2) there is one singular point Po through passes. For the cases where there is no singular which every curve rQ,Q, point, we associate with each point P of S the coordinate t(P) equal to the distance from Qo measured along in a counterclockwise direction to the end point Q p (the first one encountered) of the curve rQp,op, passing through P. Observe that we encounter the end point Q p before we reach Q 1 . We also are assuming that r is a rectifiable Jordan curve so that our surface S is simply connected. Figure 2 is representative of the surface under consideration. T h e assignment t(P) to P is unique if 0 is assigned to Qo and the distance I' from Qo to Q1measured along r in a counterclockwise sense is assigned to Q1 . For P # Qo and P # Q1 , we can uniquely assign a second coordinate s(P) to P by defining

<

< <

---f

< <

r

r.

r

s(P) = ~ Q P ~ . P ) ~ ( Q P Q , P,)

'

(7.14.2)

7.14.

257

IRREGULAR REGIONS

FIG.2

for P = Qo or P = Q1, this assignment is indeterminate, and we allow s(Qo) and s(QJ to assume any value between 0 and 1. As before, the mapping F : P -+ [t(P),s(P)]

< <

< <

maps S onto 9:0 t 1'; 0 s I . T h e map F again is not singlevalued but has a single-valued inverse F-l so that we can associate with a function f ( P ) defined on S a function f ( t , s) = f [ F - l ( t , s)] defined on 9.We can now define splines of interpolation S( f ; P ) as we did that are constant on earlier. I n this case, it is the functions in C24(94?) t = 0 and t = 1' which are relevant. and T,,,,, in the sense If we distinguish between the curves rQ,,, that they have opposite orientations, we can choose 9: 0 t 1; 0 s 1, where I denotes the length of as the image region. T h e and r,,,, are mapped by F into two line segments t = const curves rQ,,, and t' = const. We refer to t and t' if they are related in this way as conjugates and the points ( t , s) and ( t ' , 1 - s) as conjugate points. I n this terminology, the functions in C 2 ( 9 ) which are of interest are constant on t = 0, t = l', are periodic in t , and satisfy the relation f(t, s) = f(t', 1 - s). When a singular point Po is present, several modifications are required. We now take Qo as any point on I' and define t ( P )as before. I t is desirable it is to give Po the same s coordinate no matter on what curve considered to lie; consequently, we let

< <

r

< <

r,,,,;

(7.14.3.1)

for P between Q p and Po on

r,,,,;

, and (7.14.3.2)

for P between Po and Qk on rQp,4p' . If we distinguish between r,,,, and rQ,,Q, then the image is again 93': 0 t I; 0 < s < 1. The

< <

258

VII.

THE DOUBLY CUBIC SPLINE

pertinent functions in CZ4(9)are the functions periodic in t , constant along s = 8 , and satisfying f(t, s) = f(t', 1 - s) for conjugate points ( t ,s) and (t', 1 - s). Figure 3 gives a representative picture.

FIG. 3

T h e surfaces S that we have been considering need not be plane surfaces. T hus the curves I',P a , I'Q,Q, need not be plane curves. T h e mapping F : P + [t(P),s(P)] introduces a set of intrinsic coordinates into S. I t is important to note that the curves r Q , can be approximated by one-dimensional splines in the sense that their spacial coordinates can be approximated in this manner.* T h u s surfaces, defined by a finite number of points, can be obtained and can be used to approximate other smooth surfaces. Convergence can be demonstrated not only in a pointwise sense, but with respect to tangent planes and curvature as well. I n Section 7.15, we develop these ideas more fully.

r,

7.15. Surface Representation I n this section, we extend the ideas of Section 7.14 to some other classes of smooth surfaces. Three classes of surfaces are considered: ( 1 ) smooth surfaces homeomorphic to the sphere; (2) smooth surfaces homeomorphic to the torus; and (3) smooth surfaces homeomorphic to a cylinder of finite length. We consider class 1 in some depth and then indicate the modifications that must be made for the other two classes. Two different points of view are possible: we can attempt to approximate a given surface by a second surface determined by a finite number of points, or we can endeavor simply to define a surface (in a reasonable rather than a unique manner) given a finite number of points. Since the latter point of view is perhaps more basic, we adopt it. Assume that we are given N + 1 sets of points, each set consisting of * Only a finite number of curves rQ or to determine grid lines parallel to the s axis.

I'Q,Q,are

approximated; these curves serve

7.15.

259

SURFACE REPRESENTATION

coplanar points. We denote the planes by Qj(j = 0, I , ..., N ) and assume that in Qithere are mi 1 points Qi,j (i = 0, 1,..., mi)with Qo,i = Qmf,j which roughly describe a simple closed curve T i , except that To and rNdegenerate to single points. We assume the points Qo,f( j = 0, I , ..., N ) are coplanar. We think of the planes as ordered in the sense that they are parallel and that the directed distance from any plane sZj to .Qj+l is positive, although this is not a necessary condition. I n Fig. 4, we have a schematic representation.

+

Y I

X

FIG. 4

Each point Q i , j is assumed to be defined by its Cartesian coordinates (xi,j, xi,$). If d(Qi,i, Qk,j) denotes the cumulative chord length from Qi,i to Qk,i as i increases to k (k > i) and Zj denotes the total cumulative chord length, i.e., Zj = d(Qo,j,Qmj,i),we can define for each j ( j = 0, 1,..., N ) three periodic splines ,!?i(xi,j ; s), si(yi,i;s), and ,!?j(zi,j ; s) which interpolate, respectively, to the xi,j ,yi,i , and at s = si,i (i = 0, 1,..., mi),where s. . 1,3

-

4!20,9

h

Si.j>

(i = 0 , 1,..., mj ; j

= 0,

1,..., N ) .

(7.15.1)

and determine a new set of points Pi,j in each Qj by evaluating Si(xi,j ; s), ~!?~(y%,~; s), and si(xi,j; s) at so ,s1 ,..., sPk . I n particular, Qo,j= Po,i E Pzk,j ( j = 0, 1,..., N ) , Pi,o = Q0,* (i = 0, 1,..., 2k), and Pi,N Q O , N (i = 0, 1 ,..., 2k). We can in each plane form new periodic splines of interpolation to the coordinates of the Pi,j with respect to the mesh A . We denote these

260

VII.

THE DOUBLY CUBIC S P L I N E

splines by Sj(x;s), S j ( y ; s), and S j ( z ;s); they do not normally interpolate to the original coordinates, but the plane curves they define can generally be expected to resemble closely the original Ti. They have the useful property, however, that their coordinates are defined by periodic splines with respect to the common mesh A . If for fixed i and q = i + k (mod 2k) we determine a set of x coordinates by xoo = xO,N f,N+l,i

= So(x;s,),

*l,i

= 3 N . i == s N ( ' r ; si), = So(x; s p ) = xoo

= S&; Si) ,...)

= %-1(x; s,),

%I*,

% N + I , f = SN-l(x; sg),"'r

,

and similar sets of y coordinates and z coordinates, then we have 2k distinct ordered sets of coordinates. Note that the sets of points associated with i = 0 and i = 2k are identical, and sets of points associated with i and q differ only in that the ordering is reversed. We can use these sets of coordinates to define 2k oriented closed-space curves by forming t ) , and yi(z; t ) interpolating to these periodic splines q . ( x ; t ) , yi(y; coordinate values. T h e independent variable t at the points of interpolation is defined by (0 < j


where d(P,,j, Pi,l) is the cumulative chord length between Pi,j and Pi,l as the second index increases from j to 1. T h e point Qo,Nis always associated with t = by the parametrization, and the point Qo,o is associated with both t = 0 and t = 1 because of the periodicity of the splines. There is generally no value of t for which there is interpolation to the original Qij aside from these values except for i = 0, i = 2k; and i = K , where all the interpolation points are the original points Q o , j . For each i, the splines q ( x ; t ) , yi(y; t ) , and yi(z; t ) are associated with the mesh A , : 0 = ti,o < ti,l < < ti:ZN= 1 which varies with i. I n order to remove this dependence, we partition the unit interval 0 t 1 by a mesh T : 0 = to < t , < < t , = < < t,, = 1 and evaluate the splines q ( x ; t ) , K ( y ; t ) , and q ( z ; t ) for these values o f t . I n this way, we obtain 2k new sets of coordinates, and we form new periodic splines of interpolation to these quantities, but this time with respect to the common mesh T . We denote these splines by Y < , $ ( x t; ) ,

4

< <

+

7.15.

SURFACE REPRESENTATION

26 1

F T , i ( y ;t ) , Zv,i(z; t ) . Here i takes on the values 0, 1,..., 2k, and we have F&; t ) = Z , 2 k ( t ) ,Z , o ( y ; t ) = K , 2 k ( Y ; t ) , and Z,o(z; t ) = Z,&; t). Finally, we evaluate these splines for a fixed t j and form periodic splines So,i(x;s), S,,i(y; s), and S A , j ( zs); with respect to the mesh d . For eachj,

these coordinate functions define simple closed curves that are analogous 1 curves rather than to the original curves Fj except that there are 2 N N + 1, since we distinguish between orientations. T h e periodic one-dimensional splines

+

< <

define x , y , and z along the grid lines on the unit square 92:0 t 1, s 1 which are determined by the meshes T and A . A little reflection allows us to perceive that these one-dimensional splines provide sufficient and consistent information to define x , y , z as doubly periodic splines on 92 which for uniform meshes m= and d are subject to the symmetry relations

0

< <

(7.15.4)

It is only when the identifications made possible by (7.15.4) and the periodicity of the splines are made that the surface described by x(t, s), y ( t , s), and z ( t , s) is homeomorphic with the sphere. Without the identification (7.15.4), it is more like two spheres with a common north pole. and For a surface that is homeomorphic to a torus, the two curves T , are not degenerate but are identical. Consequently, the splines with respect to the t variable are naturally periodic, and the desired surface does not effectively need to be doubled in order to make them periodic. This greatly simplifies the procedure. For a surface homeomorphic to a finite cylinder, the splines with respect to the t variable are type I or type I1 splines. This causes no difficulty except that additional information must be supplied to define the requisite derivatives at the ends of the cylinder. T h e variables s and t used in these constructions can be used in situations where the surface even doubles back on itself. I n more limited situations, other variables are often more convenient. I n particular, the angle between the line joining a point on a curve Tito a fixed point and

r,,

262

VII.

THE DOUBLY CUBIC S P L I N E

a fixed line through that point can replace the s-variable in many instances. If the surface is toroidal, a similar variable can be used in place of the t variable. For a surface homeomorphic to a finite cylinder, the z coordinate itself often can be considered for the other independent variable. When it is possible to use independent variables like these, the number of times that the surface has to be redefined in order that a rectangular mesh result is significantly reduced or even eliminated. 7.16. The Surfaces of Coons

Coons [19641 has developed methods of surface representation where t b; c s d into subhe divides a rectangular region 9:a t ti; sjP1 s si (i = 1, 2 ,..., N ;j = 1, 2,..., M ) , rectangles g i i :ti-l and then he defines a surface S(t,s) over each gii which coincides along each of the four edges of gii with the restriction to that edge of one of a set of prescribed curvesfi(s) (i = 0, 1,..., N ) , g i ( t ) ( j = 0, 1,..., M ) defined on the grid lines that partition R into subrectangles Rii . This is accomplished in such a fashion that the first partial derivative of S(t, s) in the direction normal to an edge of gii[we refer to this partial derivative as the normal derivative of S(t, s)] depends only on the values of the boundary data at the ends of the edge and the values of the first derivative of the boundary data (along the edge) at the ends of the edge. I n partit i , and cular, for each gii we have S(t, siPl) = gi-l(t), tiWl t aS(t,s)/as Is=s,-l, tiWl t t i , depends only on the quantities fi-l(si-l), f i - l ( s i ) , fJ’-l(siP1), f j - l ( s i ) (and some quantities independent of the boundary data). Consequently, giPl(t) can be redefined in the interval tiPl t ti [as long as the values of gjPl(t)and giPl(t) at and ti are not altered] without affecting the normal derivative of S(t, s). I n addition, there is injected sufficient symmetry into the construction such that the normal derivative of S(t, s) is continuous across the boundary between two adjacent subrectangles. Although there is not a unique method for accomplishing these objectives, we outline one method that is very closely related to spline theory. Let gj+l(t) and g j ( t ) define S(t, s) along s = siP1 and s = si , respectively. Similarly, let fiPl(s) and fi(s) define S(t, s) along t = ti-l and t = t i , respectively. These four functions determine the values of S(t, s), aS(t, s ) / a s , and aS(t, s)/at at the vertices of gij. Finally, specify a2S(t,s ) / a s at in some fixed but arbitrary fashion at the vertices of gii . (Adjacent rectangles should have consistent values.) There is in each gii a unique double cubic s) determined by these 16 quantities.

< <

< <

< <

s(t,

< <

< <

< <

< <

7.16. THE

SURFACES OF COONS

263

Let

has the desired properties. It is not difficult to determine S(t,s). I n this regard, assume Rij is a square with unit length edges, and let

264

VII.

THE DOUBLY CUBIC SPLINE

In addition, there are prescribed

I t is straightforward from these conditions to verify that S(t, s) has the properties required of it. If it is desired to prescribe the normal derivative of S(t,s) along the boundary of quintic weight functions are needed; for higher derivatives, higher-degree polynomials are needed. T h e weight functions, however, are not restricted to polynomials. For instance, we might require S(t,s) and the analogues of the polynomial parts of (7.16.1) to satisfy the equation L*Ls = 0 for fixed t and the equation M * M s = 0 for fixed s ( L and M are linear differential operators and L* and M* their formal adjoints). If the functions fi(s) and g j ( t ) satisfy the same 2nth order differential equation L*Lh = M*Mu = 0, so that only information at the vertices of the rectangles 9ij need be specified, then the surfaces of Coons are two-dimensional splines S(t, s), where each of the one-dimensional splines obtained by restricting S(t, s) to a grid line has deficiency n at each mesh point on the grid line. Since Coons essentially considers the individual rectangles separately rather than collectively, we must admit deficiency n as indicated.

CHAPTER VIII

Generalized Splines in Two Dimensions

8.1 + Introduction

T h e material contained in this chapter is closely related to that of Chapter VII. I n this instance, however, we focus our attention on splines S,,(t, s) which in each subrectangle gii simultaneously satisfy two differential equations: one differential equation in the variable t, and the second differential equation (which may have no relation to the first) in the variable s. A second difference is that we do not require the one-dimensional splines obtained by holding one of the variables fixed to be simple splines. If we regard the fixed variable as a parameter, then all the splines in the one-parameter family of splines share a common mesh; in addition, we assume them to possess like continuity properties at interior mesh points and to satisfy end conditions of the same type. We obtain a Hilbert space theory for the splines under consideration by forming the direct product of two Hilbert spaces of one-dimensional splines. On the other hand, we obtain a characterization of splines possessing best approximation properties or minimum norm properties by establishing conditions under which the first integral relation is valid. Questions of pointwise convergence in two dimensions are reduced to similar questions in one dimension where the answers are known. T h e methods we employ generalize to higher dimensions with more or less difficulty: forming the direct product of more than two Hilbert spaces offers no difficulty; the establishment of conditions under which the first integral relation is valid, however, is beset with computational difficulties stemming primarily from the sheer length of the expressions involved. For this reason, we limit ourselves to two dimensions, where the basic method is not obscured. Although we do not consider the application of generalized two-dimensional splines to surface representation. that could be done.

265

266

VIII.

GENERALIZED SPLINES I N TWO DIMENSIONS

8.2. Basic Definition Let a rectangle 9: a ,< t ,< b; c differential operators: L,

= an(t)D:

L,

= b,(s)

*

D,"

< s < d be given along with

+ ~ , - , ( t )DY-' + ... + a,(t) ., + b,-,(s) D7-l + + b0(s). *

*

two

(8.2.1)

*

I n these differential operators, the coefficients ai(t) (i = 0, 1,..., n) possess continuous nth derivatives, and a,(t) does not vanish on [a, b]; the coefficients bj(s) ( j= 0, 1,..., m) possess continuous mth derivatives, and 6,(s) does not vanish on [c, d ] ; and D ,= d/dt, D, = d/ds. We denote by L,* and L,* the formal adjoints of L,and L, , respectively. Finally, let A , : a = t, < t , < ..- < t, = b and A , : c = so < s1 < -a'sM = d determine a mesh T which partitions 9 into N M subrectangles g i j ( i = 1,2,..., N ; j = 1 , 2,..., M ) .

Definition 8.2.1. S,(t, s) is a (two-dimensional generalized) spline on 92 with respect to Z- associated with the operators L , and L, provided (a) for each s in [c, d] S J t , s) is a heterogeneous spline on [a, 61 with respect to d , associated with the operator L , , and (b) for each t in [a, 61 S,(t, s) is a heterogeneous spline on [c, d] with respect to A , associated with the operator L, . I t is of interest at this point to examine the significance of the requirement that L,*L,S, = 0 and L,*L,S, = 0 within each rectangle g i j . From the fact that L,*L,S, = 0, we have in g i j %(t, s)

=4

, j ; s) % ( t )

+ a2(i,j; s) %(t) + + **.

a2n(i,j;

4 UznW,

where the functions ui(t) (i = 1, 2, ..., 2n) are a fundamental set of solutions of L,*L,u = 0. Consequently, L,*L,S, = 0 implies 0 = L,*L,a,(i,j; s) ul(t)

Hence, L,*L,a,(i, j ; s) ~ k ( i , j s) ;

= ckl(i,

+ L,*L,a,(i,j; s) u&) + --- + L,*L,a,,(i,j;

=

0 (K

j ) vl(s)

=

s)u&).

1, 2,..., 2n). I t follows that

+ ckz(i,j ) uz(s) + + .-*

c k , 2 A i 7 j )~ 2 r n ( s ) ,

where the functions vl(s) (I = 1, 2,..., 2m) are a fundamental set of solution of L,*L,v = 0. This establishes the representation

cc 2n

%(t, s) =

for ( t , s) in Bij .

2m

k=l 2=1

Ckdi,j)

%(t)

w

(8.2.2)

8.3.

THE FUNDAMENTAL IDENTITY

267

From the representation (8.2.2)) it follows that, if @S,(t, s ) / a t . is continuous at t = ti , then @+“S,(t,s)/as@ta (6 = 0, 1,..., 2m) is continuous at t = ti with the possible exception of the points (ti , si) ( j = 0, 1,..., M ) . A similar result holds with the roles of t and s interchanged. We also observe that (8.2.2) permits us to interchange the order of differentiation in computing the partial derivatives of S,(t, s).

8.3. The Fundamental Identity T h e bilinear concomitant P(u, v) associated with the linear differential operator L , can be expressed as (8.3.1.1)

where Pj(L, , v ; t ) =

c (- l)i j

{an+i(t)

v(t)}‘i’.

(8.3.1.2)

i=O

[In Section 6.2, we denoted pj(L,LS, ; t ) by &(S, ; t).] Since the functions we are currently considering depend on two variables t and s and there are two operators L , and L, , we employ the modified notation n-1

< <

< <

Letf(t, s) defined on a rectangle 9: a t 6; c s d be a function whose partial derivatives not involving more than n differentiations with respect to t and not more than m differentiations with respect to s are continuous on 92. Let A , : a = to < t , < -.-< t, = b and d, : c = so < s1 < -.-< sM = d define a mesh rr on 8 ; let S,(t, s) be

268

VIII.

GENERALIZED SPLINES I N TWO DIMENSIONS

a spline on 9 with respect to and consider the integral I

=

T

j:j”L,L,{f(t, a

associated with the operators L , and L, ;

s)

-

S,(t, s))L,L,S,(t,

Since L,*L,S,(t, s) = 0 and L,*L,S,(t, s) rectangle g i i ,we have

=

S)

dt ds.

0 within each sub-

(8.3.3)

If we insert I into the identity

we obtain

which is the fundamental identity for generalized splines in two dimensions. AIthough discontinuities may occur in at mesh points or along

8.4.

TYPES OF SPLINES

269

grid lines, they are simple jump discontinuities; the evaluations indicated in I must be interpreted in terms of left-hand limits and right-hand limits.

8.4. Types of Splines With respect to a m e s h d : a = to < t, < < t, = b, there are many varieties of splines on [a, b] associated with a linear differential operator L. Any particular spline Sn(t)depends on a finite number of independent parameters which we have termed defining values; s A ( t ) is the result of a particular assignment of numerical values to these parameters. If there are k parameters and each is allowed to vary over the field of real numbers, a k-parameter family FA of splines with respect to d results. T h e family FA consists of splines differing only in the assignment of numerical values to these parameters and, consequently, consists of splines possessing similar continuity properties at mesh points. I n this sense, FA defines a type of spline. Previously, for certain classes of splines, we have defined a cardinal spline as one for which all parameter values are zero except for one that is unity. I n this manner, with FAwe can associate k linearly independent splines ei(t) (i = 1, 2, ..., k), and FA is a linear space with the ei(t) as a basis. As a consequence, if S,(t) is in FA , then k

sA(t> = C aiei(t>.

(8.4.1)

i=l

Cardinal splines arise if we impose, for example, the conditions of Definition 8.4.1 or Definition 8.4.2, which follow. *

Definition 8.4.1. Let S A ( t ) be in FA. Then FA is a family of heterogeneous splines of explicit type if each ai in (8.4.1) can be identified as the value of a derivative S y ) ( t )for some a ( a = 0, 1, ..., n - 1 ) at some mesh point of d (in a manner independent of the particular spline under consideration). Moreover, if any of the quantities Sy)(ti) ( a = 0, I ,..., n - 1; i = 0, 1,..., N - 1) does not appear in this way, then ,Ba(L,L S , ; t ) is continuous at t = ti ( i = 1, 2,..., N - l), or ,Bm(L, L S ; t ) vanishes at t = ti ( i = 0, N ) . I n the earlier part of the book, we considered a number of very specific types of splines; most of these are splines of explicit type (subject to a proper choice of basis elements). T h e present definition, however

* These definitions do not cover all classes of splines; for instance, periodic splines. Modified definitions are easily formulated.

270

VIII.

GENERALIZED SPLINES I N TWO DIMENSIONS

(particularly its analog in two dimensions), greatly facilitates the statement of many basic results.

Definition 8.4.2. Let F,, be a family of two-dimensional splines on 92 with respect to n- with a basis of k splines hi(t, s) (i = 1, 2,..., k). T h e family F , consists of heterogeneous splines of explicit type if (8.4.2)

implies that each ai can be identified [in a manner independent of the particular spline S,,(t, s) under consideration] as the value of a partial y derivative aa+yS=(t,s)/at. as7 at some mesh point of x for some cy ( a = 0, 1,..., n - 1; y = 0, 1,..., m - 1). Moreover, if any of the quantities 3 + y S , ( t i , sj)/ata asY ( a = 0, 1,..., n - 1; y 70, 1,..., m - 1; i = 0, 1,..., N ; j = 0, 1,..., M ) does not appear in this manner, then &[L,, Pa(Lt,L,L,S , s; t ) , t ; s] is continuous* in either t or s at (ti , sj) (at interior mesh points), or either p,[L,, ,8,(Lt,L,L,S, , s; t ) , t ; s] vanishes or is continuous along the boundary at ( t i , sj) (at boundary mesh points that are not corner points) or /3,[L,, p,(L,, L,L,S,, , s; t ) , t ; s] vanishes at (ti , sj) (at corner points). When we say that S,,(t,s) is a spline of interpolation to f ( t , s) on n-, this asserts that S,,(t, s) = f ( t , s) at the mesh points of n-. We now require a stronger concept.

+

Definition 8.4.3. Let F,, consist of splines of explicit type, and let S,( f ; t, s) be in F,, . We say S,( f;t, s) is a spline of strong interpolation to f ( t , s) if the partial derivatives of S,( f ; t , s) indicated in Definition 8.4.2, which actually appear, interpolate to corresponding partial derivatives of f ( t , s) at the indicated mesh points of n-. 8.5. The First Integral Relation

< <

t b; Theorem 8.5.1. Let f ( t , s ) be a function dejined on 92:a c s d whose partial derivatives involving not more than n dtzerentiations with respect to t and not more than m dtflerentiations with respect to s are continuous on 92. Let n- be a mesh on 92 determined by A ,

< <

a = to < t ,

* By f(t,

sj -)

<

- 1 .

< t,

=b

the statement f(t, s) is continuous in t at (ti, sj) we imply both f(t, sj+) and are continuous if they differ.

8.6.

27 1

UNIQUENESS

and A , : c = s,

< s1 <

**.

< S,

= d,

and let S,( f;t, s) be a spline on 9 with respect to rr associated with the operators L , and L, . Then if S,( f;t , s) is a spline of strong interpolation to f ( t , s ) on r r , the jirst integral relation

1: Jl

{L,L,f(t, s))~ dt ds

=

s: s:

(LsL,S,( f;t , s))~ dt ds

is valid. Proof. From the fundamental identity and (8.3.4), we see that the theorem follows if I vanishes. Because of the special structure of S,( f ; t, s) as exhibited by the representation (8.2.2), we are justified in making the following two observations: (1) at interior mesh points, there is no contribution to I at ( t i , sj) from a summand if @+yS,( f ; t, s)/& asy interpolates to @+’f(t, s)/@ as? or &[L, , Pm(L,,L,L,S, , s; t ) , t ; s] is continuous in either t or s at (ti , sj); and (2) at boundary mesh points, there is no contribution if either P,[L,, Pa(L,,L,L,S, , s; t ) , t ; s] vanishes or is continuous there as a function on the boundary of W. This, however, is precisely what is implied by strong interpolation of S,( f ; t , s) to f ( t , s) on rr. T h e theorem follows.

8.6. Uniqueness Theorem 8.6.1, which follows, expresses the minimum norm property in the present setting. Again, it is a highly useful tool in investigating the uniqueness of splines of interpolation; again, it is a corollary of the first integral relation.

< <

< <

Theorem 8.6.1. Let 9:a t b; c s d be given, let n- be a mesh on W determined by A , : a = to < t , < < t, = b and A, : c = so < s1 < -.. < sM = d , and let F, be a family of splines of definite type. I f f ( t , s ) i s a function defined on W with the properties, (a) the partial derivatives of f ( t , s) involving not more than n -- 1 dafferentiations with respect to t and not more than m - I dafferentiations with respect to s are continuous on 92, ( b ) the partial derivatives of the restrictions of f ( t , s) to each subrectangle gijinvolving n dafferentiations with respect to t and m 9 . .

272

VIII.

GENERALIZED SPLINES I N TWO DIMENSIONS

dzflerentiations with respect to s are continuous on gij, and if S,( f;t , s) is a spline of strong interpolation to.f(t, s), then

j:s” {LsLtf(t,s)}~ dt ds 3 J: j”{L,L,S,(t, s)}~ dt ds. a

a

Given a family F , of explicit type, any solution of L,L,f = 0 is a member of F , .* I n particular, the zero function Z ( t , s) (the null spline) is a member of F, .

Definition 8.6,1. If the only spline in F , satisfying L,L,S, = 0 which is a spline of strong interpolation to Z(t, s) is Z ( t , s) itself, we say F , possesses the uniqueness property. Observe that, if the restrictions of S,(t, s) to the grid lines (these are one-dimensional splines of strong interpolation to the zero function) are identically zero, then it follows from the representation (8.2.2) that S,(t, s) is identically zero. Theorem 8.6.2. If F , has the uniqueness property and S,( f;t , s ) is a spline of strong interpolation to a function f ( t , s ) satisfying the conditions of Theorem 8.6.1, then S,( f;t , s) is unique. Proof. T h e difference of any two splines of strong interpolation to f ( t , s) is a spline of strong interpolation to Z ( t , s). From the first integral relation, it follows that the difference is a solution of L,L,S, = 0. Consequently, the difference is identically zero, and the theorem follows.

8.7. Existence Once the cardinal splines comprising a basis for a family F , of splines of definite type have been determined, it is a simple matter to construct the spline S,(f; t , s) in F, of strong interpolation to a function f ( t , s). Only certain partial derivatives of f ( t , s) need then be determined. This approach, though, assumes the existence and availability of the cardinal splines. Two-dimensional splines, however, are defined in terms of onedimensional splines. I n particular, the representation (8.3.3) is a cont b; c s d be given, and sequence of this fact. Let &?: a to < t, < < t, = b and d, : let T be determined by A , : a c = so < s1 < < sM = d. From the representation (8.3.3), it is evident that in each subrectangle gij there are 4nm coefficients cii that

< <

< <

I =

* IfLLtf

=

0,any spline of strong interpolation tof(t,

s) in

isf(t, s) itself.

8.7.

273

EXISTENCE

must be determined. If we can properly formulate a system of linear equations for determining the cii (there are a total of 4nmNM such coefficients), then we could use the uniqueness property to establish the solvability of the equations. As in earlier arguments, two solutions to the associated homogeneous system would yield two splines in F, of strong interpolation to the zero function, which would be a contradiction. I n order to obtain a proper system of equations for the cij , however, we must ensure that the conditions imposed on S,(f; t, s) so that the first integral relation is valid are satisfied. At each interior mesh point (ti, sj) there are 4nm such conditions: (a) a=+~S*(f; t , s)/& 3s. ( a = 0, 1,..., n - 1 ; y = 0, 1,..., m - 1) must be continuous at (ti, s i ) with respect to both t and s (this amounts to 3nm conditions) and (b) either 3=+~5’,(f;t, s ) / a t a as7 interpolates to aa+yf(t,s ) / a t a as? or P y [ L P&t, LLLSS,, s; t ) , t+; $1 and P , [ L P a ( ~ s , U s ~ T s;i , t ) , t - ; sl are continuous in s at ( t i , si) for 01 = 0, 1,..., n - 1; y = 0, 1 ,..., m - 1 [the roles of t and s may be interchanged in (b)]. Thus, except when condition (a) and the first alternative in condition (b) apply, there are five conditions rather than four to be satisfied at (ti , sj). Since the conditions that must hold at boundary mesh points are unaffected, we find that in some cases there are apparently more conditions than available coefficients. Consequently, we can expect that (at least formally) there may be more than 4 n m N M conditions to be imposed on the cij. I n reality, however, these conditions are not necessarily overrestrictive. For instance, suppose that FA, is a family of splines of explicit type on the interval [a, 61 and FA, is a second family of splines of explicit type, this time on the interval [c, 4. Let {hi(t)}(i = 1, 2, ..., k) be the set of cardinal splines associated with F A , , and let {gj(s)) ( j = 1 , 2, ..., 1) be the set of cardinal splines associated with FA,. Consider the linear space F,generated by the set {hi(t)gj(s))(i = 1, 2, ..., k ; j = 1, 2,.,., I ) of pairwise products. We leave it to the reader to verify that F, consists of two-dimensional splines on 92 with respect to T ; indeed, F, is a family of splines of explicit type. Observe in this regard that P Y P S

,P&

,~ , L s h , g ,, s; t ) t ; SI

=P ( L s

,Lsg, ; S) A(Lt , a

; t ) .*

(8.7.1)

T h e concept of partial splines can also be used (as in Chapter VII) to construct two-dimensional generalized splines. Again, however, the two families of splines involved in this construction should both be of

* For these splines there are, in view of (8.7.1), only four conditions to be imposed at each interior mesh point. I t can be shown that under these circumstances there are 4(M - 1)(N - 1)nm conditions from the interior mesh points and 4(N M I)nm conditions from the boundary mesh points, a total of 4NMnm conditions.

+

+

274

VIII.

GENERALIZED SPLINES I N TWO DIMENSIONS

explicit type. Under these conditions, the first integral relation prevails, and the uniqueness property will then permit the interchange of the roles of t and s as in Chapter VII.

8 3 . Convergence T h e convergence argument given in Section 7.10 applies here as well and reduces the question of convergence in two dimensions to questions of convergence in one dimFnsion. I t is required in this argument, however, that f ( t , s) and its partial derivatives have sufficient continuity to permit certain interchanges in the order of differentiation. Theorem 8.8. I , which follows, is representative of the type of convergence theorem obtainable by this means. Its proof patterns that used in Section 7.10 and is omitted. T h e rate of convergence could also be estimated, if desired, in terms of the rates of convergence of the one-dimensional splines involved.

< <

< <

Theorem 8.8.1. Let 9i?:a t b; c s d be given, together with a sequence of meshes { r k (k } = 1, 2, ...) on W determined by two sequences {A:: a = tok < tlk < - * * < t"., = b} (k = I , 2, ...) and {A:: c = sok < slk < ..-
<

uniformly in s and tik, and f o r each sik we have

uniformly in t and sjk, then

uniformly in s and t.

8.9.

275

HILBERT SPACE THEORY

8.9. Hilbert Space Theory

< <

< <

If we are given a rectangle 9: a t b; c s d and canonical mesh bases {Si(Ll; t)} (i = 1, 2,...) and {Sj(L,; s)} ( j = 1, 2 ,...) for X " ( a , b) and X m ( c , d), respectively, we can form the direct product of X n ( a , b) and Xm(c,d ) in the Hilbert space sense (cf. Sard [1963, p. 3541). We denote this direct product by Xmn(W).Moreover, Xm"(9) can be interpreted as the closure with respect to the pseudo-norm defined by

llflIZ

s"

=

a

c

IW

(8.9.1)

S f ( t , s)I2 ds dt

(here functions differing by a solution of LIL,f = 0 are identified) of the function space generated by the pairwise products of the elements of the canonical bases for X n ( u , b) and Xm(c,d). Consequently, the functions

(i = 1, 2 ,...;j

Sij(t, s) = Si(Lt ; t ) Sj(L, ; S)

=

(8.9.2)

1, 2,...)

comprise a basis for Zmn(B). We denote the space of functions f(t, s) whose a th partial derivatives involving no more than /3 differentiations with respect to t and no more than y differentiations with respect to s are continuous on 92 by C i P y ( 9 ) . In this terminology, the spaces CEtz(9) and Czh:$z)(9) are subspaces Since the basis elements (8.9.2) are in C;;t,(W), the latter of Zmn(9). is dense in Zmn(92). Iff(t, s) is in C:$r(B),then we can givef(t, s) the representation

uij =

sI

f(t, s) = g(t, s) d

e

+c

UijSij(t7

i,i

f ( t , s) L,L,Sij(t, s) ds dt

4,

(8.9.3)

(i = 1, 2,...; j

= 1,

2,...),

L,L,g(t, s ) = 0.

Moreover, we know from Theorem 8.8.1 that under mild restrictions the convergence is uniform in t and s. As a consequence, we can proceed just as in Section 5.17 and obtain the representation f ( t , s ) = g(t, s )

I

+ [ Y ( t , s; t', s') b

d

.L,L,f(t',

s') ds'

dt',

(8.9.4.1)

U " C

where Y(t,s; t', s') =

lim

N-tm

C

..

%,3
Si,i(t,s) .LtL,Si,j(t',s').

(8.9.4.2)

276

VIII.

GENERALIZED SPLINES I N TWO DIMENSIONS

T h e application of Eqs. (8.9.4) to the solution of partial differential equations of the form L,L,h(t, s) =f(t, s) (8.9.5) and the related eigenvalue problems proceeds along the same lines as in Section 6.18. T h e detracting feature is that the factorization of the partial differential operator required by (8.9.5) is not generally possible; A change of coordinates moreover, we are restricted to the rectangle 9. of the type described in Section 7.14 often can be used to remove the latter objection, but the coordinate change will generally destroy a factorization of the form (8.9.5) if it has been achieved. With respect to the approximation and representation of linear functionals, the situation is much better. One reason for the amenability of linear functionals is that they are often defined intrinsically rather than in terms of a particular coordinate system. For example, let

where D is a simply connected closed region that is starlike with respect to a point p a and d A is an element of area. Introduce into D polar coordinates r and 8 with the origin at p , . I n terms of these coordinates, let f = g(8) describe the boundary of D.I n practice, g(8) could be a simple cubic spline in the variable 8. If we now introduce new coordinates

p

= r/T,

ff

(8.9.7)

= 8;

then I

=

B

j’j 2 h 4, o

n

ar

g(.)z

ar

I

dm.

(8.9.8)

T h e preceding example illustrates how a coordinate system may be introduced into a region D such that the range of the coordinates is rectangular. Once this is done, a large class of functionals can be represented and approximated just as in Chapters V and VI. T h e differences from the one-dimensional methods offer no difficulty.

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1964e Spline functions and the problem of graduation, Proc. Natl. Acad. Sci. U . S . 52, 947-950. 1965 On monosplines of least deviation and best quadrature formulae, J. Soc. Ind. Appl. Math., Numerical Anal. Ser. 3 2, 145-170. Schoenberg, I. J., and Whitney, A. I949 Sur la positivitk des dkterminants de translations de fonctions de frkquence de P6lya avec une application au probleme d’interpolation par les fonctions “spline,” Compt. Rend. 228, 1996-1998. 1953 On Polya frequency functions. 111, Trans. Am. Math. SOC.74, 246-259. Schweikert, D. G. 1966 An interpolation curve using a spline in tension, J . Math. Phys. (to appear). Secrest, D. 1965 Best integration formulas and best error bound’s, Math. Computation 19, 79-83. 1965a Error bounds for interpolation and differentiation by the use of spline functions, J . Sac. Ind. Appl. Math., Numerical Anal. Ser. B 2, 440-447. Seifert, H. S. (ed.) 1959 “Space Technology.” Wiley, New York. Sharma, A., and Meir, A. abs. 1964 Convergence of spline functions, Notices Am. Math. Sac. 64T-496. 1966 Degree of approximation of spline interpolation, J . Math. Mech. 15,759-767. Sokolnikoff, I. S. 1956 “Mathematical Theory of Elasticity.” McGraw-Hill, New York. Taylor, A. E. 1958 “Introduction to Functional Analysis.” Wiley, New York. Theilheimer, F., and Starkweather, W. 1961 T h e fairing of ship lines on a high-speed computer, Numerical Tables Aids Computation 15, 338-355. Todd, J. (ed.) 1962 “Survey of Numerical Analysis.” McGraw-Hill, New York. Walsh, J. L., Ahlberg, J. H., and Nilson, E. N. 1962 Best approximation properties of the spline fit, J . Math. Mech. 11, 225-234. Ziegler, Z. abs. 1965 On the convergence of nth order spline functions, Notices Am. Math. SOC. 65T-354.

INDEX Ahlberg, J. H., 2, 3, 4, 5, 6, 7, 70. 78, 136, 140, 142, 237 Atteia, M., 6 Auslender, S., 143 Beamfit see cubic spline Beam theory, 1 , 3, 77 Bellman, R. E., 56 Best approximation property, 4 for cubic splines, 16-19, 77-78 for doubly cubic splines, 244-245 for generalized splines, 200-201 for polynomial splines, 157-1 59 Bicubic splines, 5-6 see also doubly cubic splines Birkhoff, G., 4, 5 , 29, 237

C;,J") definition of, 275 Canonical mesh bases for cubic splines, 101-103 for doubly cubic splines, 249 for generalized splines, 219-220, 275 for polynomial splines, 179-1 82 Cardinal splines, 52-58, 245-247, 25 1-254 Circulant matrix, 36, 133, 148-149 Convergence in norm for cubic splines, 98-101 for doubly cubic splines, 247-248 for generalized splines, 214-219 for polynomial splines, 176-179 Convergence properties, 4-5 for cubic splines, 19-34, 61-74, 87-96 for doubly cubic splines, 247-248 for generalized splines, 201-212, 274 for polynomial splines, 135-143, 148152, 16G-174 see also convergence in norm Coons surfaces, 262-264 Cubic splines 1, 9-108 best approximation property, 16-19, 77-78 canonical mesh bases, 101-103 cardinal splines, 52-58

convergence, 19-34, 61-74, 87-96 convergence in norm, 98-10] curve fitting, 50-52 deficiency of, 24 defining values, 82, 97 end conditions, 11, 13-14, 50 equal intervals, 9, 34-42 equations for, 10-16, 84-87 existence, 16-19, 61-74, 84 first integral relation, 77-82 fundamental identity, 78-79 Hilbert space theory of, 97-107 integral equations, 57-59 linear functionals, 103-107 minimum norm property, 75-77 orthogonality, 97-1 01 parametric splines, 51 periodic cubic splines, 10 second integral relation, 89-91 simple cubic splines, 78 spline-on-a-spline, 44, 48-50 type I, 75 type 1', 75 type 11, 75 type 11', 75 uniqueness, 16-19, 82-83 Cumulative chord length, 51, 254-262 Curve fitting 1-2, 50-52, 143 see also surfaces Dahlquist, G., 233 Davis, P. J., 19, 220 de Boor, C., 4, 5, 6, 7, 29, 237 Deficiency of a spline, 7 for cubic splines, 24 for generalized splines, 191-1 92 for polynomial splines, 123, 143-147, 157, 163-164 for quintic splines, 123, 143-148 Defining values, 82, 97, 175, 235 Degree of a spline, 109 Differential equations, 52-57, 228-233 see also partial differential equations Differentiation, see numerical differentiation

28 1

282

INDEX

Direct products (of Hilbert spaces) 249 Doubly cubic splines, 5, 235-264 best approximation property, 244-245 canonical mesh bases, 249 cardinal splines, 245-247, 251-254 convergence, 247-248 convergence in norm, 250 defining values, 235 doubly periodic splines, 236 existence, 239, 243-244 first integral relation, 242 fundamental identity, 240-242 Hilbert space theory, 249-251 irregular regions, 254-258 %;(a),275 minimum norm property, 242-243 partial differential equations, 251-254 partial splines, 237-239 second integral relation, 248 simple doubly cubic splines, 235 surfaces, 256-262 type I, 235-236 type 1’, 236 type 11, 236 type 11’, 236 uniqueness, 239, 243-244 see also generalized splines (two-dimensional) Doubly periodic splines, 236 Eigenvalue problem, 57-58, 228-230 Elastica, 1, 3 End conditions cubic splines, 11, 13-14, 50 curve fitting, 50 modified type k, 170 polynomial splines, 122-123, 158 type I, 75, 113, 193-194, 235-236 type 1’, 75, 113, 193-194, 236 type 11, 75, 113, 193-194, 236 type 11’, 75, 113, 193-194, 236 type k, 167-168, 194 see also periodic splines Equal intervals cubic splines, 9, 34-42 polynomial splines, 124-135, 148-152 see also uniform meshes Existence of splines, 2 of cubic splines, 16-19, 61-74, 84 of doubly cubic splines, 239, 243-244

of generalized splines, 199-200, 272-274 of polynomial splines, 132-1 35, 165-1 66 Explicit type, 269-270 Fejbr, L., 23-24, 147-148 First integral relation, 3 for cubic splines, 77-82 for doubly cubic splines, 242 for generalized splines, 193-195,270,271 for polynomial splines, 155-1 56 Fundamental identiy for cubic splines, 78-79 for doubly cubic splines, 240-242 for generalized splines, 192-193,267-269 for polynomial splines, 154-155 Garabedian, H., 5, 237 Generalized splines, 6, 191-233, 265-276 best approximation property, 200-201 canonical mesh bases, 219-220, 275 convergence, 201-212, 274 convergence in norm, 214-219 deficiency of, 191-192 differential equations, 228-233 equations for, 197-199 existence, 199-200, 272-274 explicit type, 269-270 first integral relation, 193-195, 270-271 fundamental identity, 192-1 93, 267-269 heterogeneous splines, 194 Hilbert space theory, 213-233, 275-276 integral equations, 230-232 linear functionals, 220-233, 275-276 minimum norm property, 195-196, 271-272 numerical integration, 224-228 order of, 191 orthogonality, 21 8 partial differential equations, 276 partial splines, 273 Peano kernels, 220-224 periodic splines, 194 second integral relation, 204-206 simple splines, 194 strong interpolation, 270 two-dimensional, 265-276 type I, 193-194 type 1’, 193-194 type 11, 193-194 type 11’, 193-194

INDEX

type k, 194 uniqueness, 196-197, 271-272 uniqueness property, 272 Gershgorin’s theorem, 16 Golomb, M., 7-8 Greville, T. N. E., 4, 6, 105 Heterogeneous splines, 7, 194-196 Hilbert space theory, 3-4 for cubic splines, 97-107 for doubly cubic splines, 249-251 for generalized splines, 21 3-233,275-276 for polynomial splines, 174-190 Hille, E., 134 Holladay, J. C., 3, 8, 46, 75, 77, 78, 80 Integral equations, 57-61, 230-232 Integration, see numerical integration Irregular regions, 254-258

x%,b)

definition of, 75 .G(9) definition of, 275 x%,b ) definition of, 75 Kalaba, R. E., 56 Limits on convergence, 95-96, 174, 211212 Linear functionals, 6-7, 103-107, 185-189, 220-233, 275-276 Lynch, R. E., 4, 6, 7 Mathematical spline, 1, 9 Meir, A., 4, 27 Mesh bases, 101-103, 179-1 82 see also canonical mesh bases Mesh restrictions, 22, 25, 94-95 Method of cardinal splines, see cardinal splines Minimum curvature property, 2, 3 see minimum norm property Minimum norm property, 3 for cubic splines, 75-77 for doubly cubic splines, 242-243 for generalized splines, 195-196,271-272 for polynomial splines, 156-157 Modified type k, 170 see also type k

283

Multidimensional splines, 5 , 6 see also two-dimensional splines Nilson, E. N., 2, 3, 4, 5 , 6, 7, 70, 78, 136, 140, 142, 237 Nonlinear differential equations, 56 Numerical differentiation, 42-57 Numerical integration, 42-50, 145-146, 224-228 Order of a spline, 191 Orthogonality, cubic splines, 97-101 generalized splines, 21 3-219 polynomial splines, 174-179 see also canonical mesh bases Parametric splines, 51, 254-258 surfaces, 258-262 Partial differential equations numerical solution of, 251-254, 276 Partial splines, 237-239, 273 Peano, G., 105, 185 Peano kernel, 6,103-105,182-189,220-224 Periodic splines, 10, 148-152, 156, 167, 194, 236 Polynomial splines, 109-1 89 best approximation property, 157-1 59 canonical mesh bases, 179-182 convergence, 135-143, 148-152, 166-174 convergence in norm, 176-179 deficiency of, 123,143-148, 157,163-164 defining values, 175 degree of, 109 end conditions, 122-123, 158 equal intervals, 124-135, 148-152 equations for, 109-123, 160-165 even degree, 109-1 10, 153 existence of, 132-135, 165-166 first integral relation, 155-156 fundamental identity, 154-1 55 Hilbert space theory, 174189 limits on convergence, 174 linear functionals, 185-189 minimum norm property, 156-1 57 modified type k, 170 numerical integration, 145-146 odd degree, 5 , 153-189 orthogonality, 174-179 Peano kernels, 182-1 89

INDEX

periodic splines, 148-152, 156, 167 quintic splines, 143-148 remainder formulas, 185-189 second integral relation, 168-170 simple splines, I13 type I, 113 type I’, 113 type 11, 113 type 11’, 113 type k, 167-168 uniform mesh, 148-152 uniqueness, 159-160 Quadrature, see numerical integration Quintic splines, 143-148 Remainder formulas, 103-105, 185-189, 221-228 Sard, A., 6, 105, 185,222,227,233,275 Schoenberg, I. J., 2, 4, 5, 6, 7, 44, 75, 106, 185 Schweikert, D. G., 6 Second integral relation, 5 for cubic splines, 89-91 for double cubic splines, 248 for generalized splines, 204-206 for polynomial splines, 168-170 Secrest, D., 8 Sharma, A,, 4, 27 Simple splines, 7, 78, 113, 194, 235 Sokolnikoff, I. S., 2, 3, 77 Space technology, 107-108 Spline equations for cubic splines, 10-16, 84-87 for generalized splines, 197-1 99

for polynomial splines, 109-123, 160-165 Splines in tension, 6 Splines of even degree, 109-1 10, 153 Splines of odd degree, 153-189 see also polynomial splines, cubic splines Spline-on-spline, 44, 48-50 Spline surfaces, 256-262 see also Coons surfaces Strong interpolation, 270 Surfaces, see spline surfaces, Coons surfaces Two-dimensional splines doubly cubic splines, 235-264 generalized splines, 265-276 Type I, 75, 113, 193-194, 235-236 Type 1’, 75, 113, 193-194, 236 Type 11, 75, 113, 193-194, 236 Type 11’, 75, 113, 193-194, 236 Type k, 167-168, 170, 194 Uniform mesh, 148-152 see also equal intervals Uniqueness of cubic splines, 16-19, 82-83 of doubly cubic splines, 239, 243-244 of generalized splines, 196-197, 271-272 of polynomial splines, 159-160 Uniqueness property, 272 Walsh, J. I,., 2, 3, 4, 5 , 6, 7, 78, 136, 140, 142, 237 Weinberger, H. F., 7-8 Whitney, A,, 2 Ziegler, Z., 5 , 136