Realizability theory for continuous linear systems, Volume 97 (Mathematics in Science and Engineering)

Realizability Theory for Continuous Linear Systems

This is Volume 97 in MATHEMATICS IN SCIENCE AND ENGINEERING A series of monographs and textbooks Edited by RICHARD BELLMAN, University of Southern California The complete listing of books in this series is available from the Publisher upon request.

Realizability Theory for Continuous Linear Systems A. H. Zemanian Department of Applied Mathematics and Statistics State University of New York at Stony Brook Stony Brook, New York

ACADEMIC PRESS

New York and London

1972

COPYRIQHT 0 1972, BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED. N O PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER.

ACADEMIC PRESS, INC.

I 1 1 Fifth Avenue, New York. New York 10003

United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. 24/28 Oval Road, London N W 1

LIBRARY OF CONGRESS CATALOG CARDNUMBER:12 - 11345 AMS (MOS) 1970 Subject Classification: 93A05 PRINTED IN THE UNITED STATES OF AMERICA

To the Memory of M Y

F A T H E R

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Contents

Preface ................................................................ Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi xv

CHAPTER 1 . Vector-Valued Functions Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........... 1.2. Notations and Terminology .............................. ........... 1.3. Continuous Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........... 1.4. Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5. Repeated Integration and Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. ........... 1.6. Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I .7. Banach-Space-Valued Analytic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........... I .8. Contour Integration ...................................... 1.1.

1 2 5

6 9 12 17

20

CHAPTER 2 . Integration with Vector-Valued Functions and Operator-Valued Measures 2.1. lntroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Operator-Valued Measures ............................... 2.3. a-Finite Operator-Valued Measures ........................ vii

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

23 23 29

viii 2.4. 2.5. 2.6.

CONTENTS

Tensor Products and Vector-Valued Functions .......................... Integration of Vector-Valued Functions .................................. Sesquilinear Forms Generated by PO Measures ..........................

34 38 43

CHAPTER 3 . Banach-Space-Valued Testing Functions and Distributions 3.1. Introduction . . . . . . . . . ............................................ 3.2. The Basic Testing-Func Spacearn(A) ............................... 3.3. Distributions . . . . . . . . . . . . . . . . . .................................... 3.4. Local Structure . . . . . . . . . . . . . . . .................................... 3.5. The Correspondence between [ 9 ( A ) ; B ] and [g; [A; B ] ]. . . . . . . . . . . . . . . . . . . 3.6. The p-Type Testing Function Spaces . . .... .. 3.7. Generalized Functions ................................................. 3.8. &-Type Testing Functions and Distributions ............................

49 50 52 57 61 64 67 72

CHAPTER 4. Kernel Operators Introduction ......................................................... Systems and Operators ................................................ 4.3. The Space 8 = 9 ( Y ) ................................................ 4.4. The Kernel Theorem ................................................. 4.5. Kernel Operators ..................................................... 4.6. Causality and Kernel Operators ........................................

4.1. 4.2.

76 77 81 85 89 93

CHAPTER 5 . Convolution Operators

.............. ............. Introduction . . . . . . . . . Convolution . . . . . . . . . ................................... ..................................... Special Cases . . . . . . . . . . . . . . perators with Shifting and 5.4. The Commutativity of Differentiation ..................................................... 5.5. Regularization . . . . . . . . . . . . ........................ 5.6. Primitives....... .................. ........................... 5.7. Direct Products ...................................................... 5.8. Distributions That Are Independent of Certain Coordinates . . . 5.9. A Change-of-Variable Formu ........................................ 5.10. Convolution Operators . . . . . ........................................ 5.1 1. Causality and Convolution Operators .................................. 5.1.

5.2. 5.3.

96 97 98

104 108 110

111 112 115

CHAPTER 6 . The Laplace Transformation 6.1. 6.2.

Introduction ......................................................... The Definition of the Laplace Transformation ............................

117 117

ix

CONTENTS

.

6.3. Analyticity and the Exchange Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , . . 119 6.4. Inversion and Uniqueness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 6.5. A Causality Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

.

.

CHAPTER 7. The Scattering Formulism 7.1. 7.2. 7.3. 7.4. 7.5. 7.6. 7.1. 7.8.

Introduction ......................................................... Preliminary Considerations Concerning L,-Type Distributions , . . . . , . . . . . . . . Scatter-Passivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bounded* Scattering Transforms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Realizability of Bounded* Scattering Transforms . . . . . . . . . . . . . . . . . . . . Bounded*-Real Scattering Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lossless Hilbert Ports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Lossless Hilbert n-Port . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

124 125 128 131 134 137 139 143

CHAPTER 8. The Admittance Formulism 8.1. 8.2. 8.3. 8.4. 8.5. 8.6. 8.7. 8.8. 8.9. 8.10. 8.1 1. 8.12. 8.13. 8.14. 8.15. 8.16.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Passivity . .. . . . . . . .. . . .. .. .. . . .. .. .... . . .. .. . Linearity and Semipassivity Imply Continuity . . . . The Fourier Transformation on Y ( H ) . . . . . . . . . . . . . . . ............... Local Mappings.. . . . . . . . . . . . . . . . . . . . . . . . . . . . Positive Sesquilinear Forms on 9 x 9 .................. Positive Sesquilinear Forms on 9 ( H ) x Certain Semipassive Mappings of 9 ( H ) into & H ) . An Extension of the Bochner-Schwartz Theorem . ............. Representations for Certain Causal Semipassive M ........... A Representation for Positive* Transforms. . . . . . . Positive* Admittance Transforms . . . . . . . . . . . . . . . Positive* Real Admittance Transforms . . . . . . . . A Connection between Passivity and Semipassivity ......... A Connection between the Admittance and Scattering Formulisms . . . . . . . . . . The Admittance Transform of a Lossless Hilbert Port . . . . . . . . . . . . . . . . . . . . .

.

149

155 160

174

187 189 192

APPENDIX A. Linear Spaces

194

APPENDIX B. Topological Spaces

198

APPENDIX C. Topological Linear Spaces

201

D. Continuous Linear Mappings APPENDIX

206

APPENDIX E. Inductive-Limit Spaces

21 1

X

CONTENTS

APPENDIX F. Bilinear Mappings and Tensor Products

213

APPENDIX G. The Bochner Integral

216

References

222

.

Index of Symbols.. . . . . . . . . . . . . . . . . . . . . . . . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 Index .................................................................... 221

“Realizability theory” is a part of mathematical systems theory and is concerned with the following ideas. Any physical system defines a relation between the stimuli imposed on the system and the corresponding responses. Moreover, any such system is always causal and may possess other properties such as time-invariance and passivity. Two questions: How are the physical properties of the system reflected in various analytic descriptions of the relation? Conversely, given an analytic description of a relation, does there exist a corresponding physical system possessing certain specified properties ? If the latter is true, the analytic description is said to be realizable. Considerations of this sort arise in a number of physical sciences. For example, see McMillan (1952), Newcomb (1966), or Wohlers (1969) for electrical networks, Toll (1956) or Wu (1954) for scattering phenomena, and Gross (1953), Love (1956), or Meixner (1954) for viscoelasticity. This book is an exposition of realizability theory as applied to the operators generated by physical systems as mappings of stimuli into responses. This constitutes the so-called “ black box ” approach since we do not concern ourselves with the internal structure of the system at hand. Physical characteristics such as linearity, causality, time-invariance, and passivity are defined as mathematical restrictions on a given operator. Then, the two questions are xi

xii

PREFACE

answered by obtaining a description of the operator in the form of a kernel or convolution representation and establishing a variety of necessary and sufficient conditions for that representation to possess the indicated properties. Thus, the present work is an abstraction of classical realizability theory in the following way. A given representation is realized not by a physical system but rather by an operator possessing mathematically defined properties, such as causality and passivity, which have physical significance. We may state this in another way. Our primary concern is the study of physical properties and their mathematical characterizations and not the design of particular systems. Two properties we shall always impose on any operator under consideration are linearity and continuity. They are quite commonly (but by no means always) possessed by physical systems. Of course, continuity only has a meaning with respect to the topologies of the domain and range spaces of the operator. We can in general take into account a wider class of continuous linear operators by choosing a smaller domain space with a stronger topology and a larger range space with a weaker topology. With this as our motivation, we choose the basic testing-function space of distribution theory as the domain for our operators and the space of distributions as the range space. The imposition of other properties upon the operator will in general allow us to extend the operator onto wider domains in a continuous fashion. For example, time-invariance implies that the operator has a convolution representation and can therefore be extended onto the space of all distributions with compact supports. This distributional setting also provides the following facility. It allows us to obtain certain results, such as Schwartz’s kernel theorem, which simply do not hold under any formulation that permits the use of only ordinary functions. Thus, distribution theory provides a natural language for the realizability theory of continuous linear systems. Still another facet of this book should be mentioned. Almost all the realizability theories for electrical systems deal with signals that take their instantaneous values in n-dimensional Euclidean space. However, there are many systems whose signals have instantaneous values in a Hilbert or Banach space. Section 4.2 gives an example of this. For this reason, we assume that the domain and range spaces for the operator at hand consist of Banach-spacevalued distributions. Many of the results of earlier realizability theories readily carry over to this more general setting, other results go over but with difficulty, and some do not generalize at all. Moreover, the theory of Banach-spacevalued distributions is somewhat more complicated than that of scalar distributions; Chapter 3 presents an exposition of it. Still other analytical tools we shall need as a consequence of our use of Banach-space-valued distributions are the elementary calculus of functions taking their values in locally convex spaces, which is given in Chapter 1, and Hackenbroch’s

PREFACE

xiii

theory for the integration of Banach-space-valued functions with respect to operator-valued measures, a subject we discuss in Chapter 2. The systems theory in this book occurs in Chapters4,5,7, and 8. Chapter 4 is a development of Schwartz’s kernel theorem in the present context and ends with a kernel representation for our continuous linear operators. Causality appears as a support condition on the kernel. How time-invariance converts a kernel operator into a convolution operator is indicated in Chapter 5. We digress in Chapter 6 to develop those properties of the Laplace transformation that will be needed in our subsequent frequency-domain discussions. Passivity is a very strong assumption; it is from this that we get the richest realizability theory. Chapter 7 imposes a passivity condition that is appropriate for scattering phenomena, whereas a passivity condition that is suitable for an admittance formulism is exploited in Chapter 8. It is assumed that the reader is familiar with the material found in the customary undergraduate courses on advanced calculus, Lebesgue integration, and functions of a complex variable. Furthermore, a variety of standard results concerning topological linear spaces and the Bochner integral will be used. In order to make this book accessible to readers who may be unfamiliar with either of these topics, a survey of them is given in the appendixes. Although no proofs are presented, enough definitions and discussions are given to make what is presented there understandable, it is hoped, to someone with no knowledge of either subject. Almost every result concerning the aforementioned two topics that is used in this book can be found in the appendixes, and a reference to the particular appendix where it occurs is usually given. For the few remaining results of this nature that are employed, we provide references to the literature. The problems usually ask the reader either to supply the proofs of certain assertions that were made but not proved in the text or to extend the theory in various ways. On occasion, we employ a result that was stated only in a previous problem. For this reason, it is advisable for the reader to pay some attention to the problems. All theorems, corollaries, lemmas, examples, and figures are triplenumbered ; the first two numbers coincide with the corresponding section numbers. On the other hand, equations are single-numbered starting with (1) in each section.

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Acknowledgments

This book was conceived while I was a Research Fellow at the Mathematical Institute of the University of Edinburgh during the 1968-1969 academic year. My tenure in that post was supported by a grant to Professor A. Erdelyi from the Science Research Council of Great Britain. It is a pleasure to express my gratitude for that support. The subsequent development of this book was assisted by Grants GP-7577, GP-18060, and GP-27958 from the Applied Mathematics Division of the National Science Foundation under the administration of Dr. B. R. Agins. I also wish to express my gratitude to R. K. Bose, V. Dolezal, and W. Hackenbroch for various suggestions that have been incorporated into the text. Finally and once again, to my wife, thanks.

xv

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Chapter 1

Vector-Valued Functions

1.1. INTRODUCTION

The purpose of this first chapter is to present a number of results concerning the calculus of functions that map n-dimensional real Euclidean 'space into a locally convex space. In addition, the theory of analytic functions from the complex plane into a Banach space is also introduced. The discussion is quite similar to the elementary calculus and the theory of complex-valued analytic functions. Complications arise at various points, however, because of the more involved structure of locally convex and Banach spaces. Nevertheless, there are no surprises. The primary results of Riemann integration, differentiation, and functions of a complex variable carry over to the present context. The reader may, if he wishes, go directly to Chapter 2 after a perusal of the next section and refer back to the present chapter as the occasion demands. As was mentioned in the Preface, certain facts concerning topological linear spaces will be used. A compilation of them can be found in the appendixes of this book. 1

2

1. VECTOR-VALUED FUNCTIONS

1.2. NOTATIONS AND TERMINOLOGY

The present section is devoted to a description of some of the notations and terminology that are used throughout this book; almost all of them adhere to customary usage. The Index of Symbols at the end of the book lists our frequently used symbols and indicates the pages on which they are defined. Let @ be a set. The notation {4 E 0: P ( 4 ) } , or simply {4: P($)} if 0 is understood, denotes the set of all 4 E @ for which the proposition P ( 4 ) concerning 4 is true. {4i}is,is a collection of indexed elements where the index i traverses the set 1. A sequence is denoted by {+k}F= or {&: k = 1, 2, . . .}, a finite collection by {4k}:=1or {41,.. ., 4"}, and a set containing a single element 4 by {4}. We also use the abbreviated notation (4,) when it is clear what type of set we are dealing with. Let R and A be subsets of @. The notation 4, ll/, 0, . . . E R means that all the elements 4, ll/, 0, . . . are members of R ; 4, ll/, 0, . . . 4 R is the corresponding negation. If 0 is a complex linear space (see Appendix A), R + A (or R - A) denotes the set of elements of the form ll/ + 6 (or respectively ll/ - 0) where Ic/ E R and 0 E A . Similarly, if 4 is a fixed member of @ and a is a complex number, R 4 (or a n ) is the set of all elements of the form II/ 4 (or respectively all/), where I) E 0.On the other hand, R\A is the set of all elements in R that are not in A. Thus, @\A is the complement of A in 0. We also have the customary union u, intersection n, and inclusion symbols c or 3 . Furthermore, 0,and R, denote unions and intersections, respectively, over sequences {Rk} of sets in @. A similar notation is used for finite collections and indexed collections of sets, for example, (JiZlR, and n i E I R i . If @ is a topological space (Appendix B), fi is the interior of 0,and I2 the closure of 0.However, for any complex number a, 5 is the complex conjugate of a. Now, let O k 9 where k = I , . . . , n, be sets. The Cartesiunproduct @, x ... x a,, is the set of all ordered n-tuples {$J~,. . . , 4,,}, where 4, E @k and k = 1. . . . , n . A rulefthat assigns one or more elements ll/ in a set Y to each element 4 i n another set @ is called a relation. Thus, f determines a subset of @ x Y called the graph qf.f and consisting of all ordered pairs {b, I )}, where ll/ is an element assigned to 4 by f. We also use the notation f: 0 c, Y as well as f:4 Hll/ to denote the rule, the sets for which the rule is defined, and the typical elements related by$ The domain off is the set of all elements 4 on which f is defined, in this case, @; 4 is called the independent variable. The range of'fis the subset of Y consisting of all elements ll/ that are assigned by f t o members of 0 ;ll/ is called the dependent rariable.

+

+

1.2

NOTATIONS AND TERMINOLOGY

3

A function f is a relation f : @ -+ Y that assigns precisely one element in Y to each member of @; in other words,fis a function if and only if the equations $ = f 4 and 8 = fi imply that $ = 8. Synonymous with the word “function ” are the terms operator, mapping, and transformation. We say that f:4 H $ maps, carries, or transforms 4 into Y and that f is a function on (orfrom) @ into Y or a mapping of @ into Y. We also say thatfis Y-valued; when Y is the real line or the complex plane, the phrase “Y-valued” is replaced by real-valued or complex-valued, respectively. If R is a subset of Q,, the symbol f(R) denotes the set {$ E Y : $ = f i , 4 E R}. The function g that is defined only on R but coincides withfon R is called the restriction of f to R. On the other hand, f is called an extension of g. A function f is said to be one-to-one or injective and is also called an injection if the equations f 4 = $ and fx = $ imply that 4 = x. In this case, : $ -4, which maps the range off into 0; f is we have the function f called the inverse off. A function f : @ -+ Y is said to be onto or surjective and is also called a surjection if the range o f f coincides with Y . Thus, f is a surjection onto Q, whenever f is injective. I f f : @ c, Y is both injective and surjective, it is said to be bijectice or a bijection. We denote the elements in the range o f f by the alternative notations $ = f& = f (4) = {f,4). On occasion, it will be convenient to violate this symbolism by using f ( 4 ) to denote the function f rather than its range value, as was commonly done in classical mathematics. Whenever we do so, it will be clear from the context what is meant. On still other occasions when the symbol for f is rather complicated, we may use the dot notation f (.) = (f, .) in order to indicate where the independent variable should appear. For example, when t E R , we may denote the function t H cos(sin t) by cos(sin .). When -Y and W are topological linear spaces (see Appendix D and especially Sections D8 and D1 l), the symbol [ Y ;W ] denotes the linear space of all continuous linear mappings of Y into W . Unless the opposite is explicitly stated, it is always understood that [ Y ;W ] is equipped with its bounded topology. When it has the pointwise topology, it is denoted by [ Y ;W],. Thus, if A and B are Banach spaces, [ A ; B] possesses the uniform operator topology, whereas [ A ; B]“ has the strong operator topology. R“ and C” denote, respectively, the real and complex n-dimensional Euclidean spaces. Thus, an arbitrary point t E R” (or t E C“) IS an ordered n-tuple t = {tk}iZ1of real (respectively complex) numbers tk whose magnitude is

-’

We set R = R‘ and C = C ’ . R , denotes the positive half-line {t E R : 0 < t < 03). R,“ is the set of all ordered n-tuples each of whose components

1.

4

VECTOR-VALUED FUNCTIONS

is a real number or co; for example, (2, 03, - 1) is such a triplet. As before, we set Re = R t . We do not allow - 00 as a possible component of any t E R,". Thus, if a E R,", - a is an n-tuple whose components are either real numbers or - 00. If x E R e , the symbol [XI will denote the n-tuple in R," each of whose components is x ; however, the n-tuple [O] is denoted simply by 0. The number n of components in [XI will be implied by the context in which the symbol [ x ] is used. An integer k E R" is an n-tuple all of whose components are integers. A compact set in R" is a closed bounded set in R". Given any set R in R", the diameter of R is denoted by diam R and defined by diam R = sup{ I t - X I : t , x

E

R}.

If x = {xk};=1 and t = {tk};!l are members of R" or R,", the notations x It and x t mean that xk Itk and respectively xk < t k for k = 1, . . . , n. If t 2 0 (or if t > 0), t is said to be nonnegatioe (respectively positive). An interval in R" is the Cartesian product of n intervals in R. As special cases,

-=

we have ( x , y ) = { t E R": x < t < y } , ( x , y ] = { t E R": x


I y},

[ x , y ) = { t R": ~ x I t < y}, [x, y ] = {t

E R": x

I t Iy } ,

-x, y

E R,",

-X

E

(1)

R,", y E R",

x

ER",

x, y

E R".

YER,",

(2)

(3) (4)

Equation (1) is an open interval, whereas (4) is a closed interval. The symbol ( x , y ) is the same as that for the inner product in a Hilbert space, but they can be distinguished by the way they are used. The volume of any interval I in R" with endpoints x and y, where x Iy, is denoted by vol I and defined by

n n

vol I

k= 1

(yk - xk).

Let us mention a few other customary symbols. Iff is a continuous function on R" or C", its support is the closure of the set of points t for which f( 1 ) # 0 and is denoted by suppf. (The support of a distribution on R", which is defined subsequently, is also denoted by suppJ) The symbols sup and inf denote, respectively, the operations of taking the supremum and infimum of a subset of R , whereas max and min denote, respectively, the taking of the maximum and minimum of a finite subset of R. Also, lim and lim symbolize the limit superior and limit inferior, respectively. Two unusual symbols are the following. Whenever we wish to emphasize that a particular equation is a definition, we replace the equality sign by A. The notation 0 denotes the end of a proof or example. Throughout this book, A and B will denote complex Banach spaces, whereas H will be a complex Hilbert space.

1.3

5

CONTINUOUS FUNCTIONS

1.3. CONTINUOUS FUNCTIONS

In the rest of this chapter, Y always denotes a separated sequentially complete locally convex space, and a generating family of seminorms for the topology of Y . Also J is a subset of R" equipped with the topology induced by R". Letfbe a mapping on J into Y .The continuity offis defined in Appendix B4. Since J is a metric space, the continuity offis equivalent to its sequential continuity (Appendix B8). The mapping f is said to be uniformly continuous on J if, for every neighborhood R of 0 in Y ,there exists an r E R, such that the conditions t , X E J and ( t - X I < r imply that f ( t ) - f ( x ) E R. This condition is equivalent to the following. Given any E E R, and any y E r, there exists an r E R, such that the conditions t , x E J and It - x ( < r imply that y [ f ( t ) - f ( x ) ] < E . The mapping f is said to be bounded on J iff(J) is a bounded set in Y (see Appendix C8). Theorem 1.3-1. Let J be a compact subset of R", and let f be a continuous mapping of J into Y .Thenf is uniformly continuous and bounded on J . PROOF.We first establish the uniform continuity off. Since Y is locally convex, any neighborhood of 0 in Y contains a balanced convex neighborhood R of 0 in Y . Set A = +R. By the convexity of R, we have that A A c R. The continuity off implies that, for each t E J , there exists a neighborhood N ( t , r ) & { X E J : Ix - tI < r } such that

+

f (4E A + f (0

(1)

whenever x E N(t, r ) . Now, the collection of neighborhoods { N ( t , r/2)},,J covers J. But J is compact, and therefore a finite subset of this collection also covers J. That is, J c N ( t , , r 1 / 2 )u ... u N ( t , , rm/2). Next, set q

= mink rk/2,

and consider any two points t , x

I t - X I < q . There exists some N(tk, rk.2) containing t . Hence, f(t) -f(tk)EA.

E

J such that (2)

Moreover, (X-tkI

Consequently, x balanced,

E

< (x-21

+ It-tkI


N(tk, rk). By (I), f ( x ) - f ( t k ) E f(tk)

- f ( x ) A*

A.

Therefore, since A is

(3)

6

1. VECTOR-VALUED FUNCTIONS

We can combine (2) and (3) to write f ( t ) - f(x) = f ( t ) - f ( t k )

+

f(fk)

- f ( x ) E A -I- A

0,

This proves the uniform continuity off on J . To show its boundedness, let y be any member of r. Since both y and f are continuous functions, so too is the mapping t ~ + y [ f ( t ) ] of J into R . Hence, y[f(J)] is bounded in R . Since y is arbitrary, this implies that f ( J ) is bounded in Y . 0 1.4. INTEGRATION

Integrals of continuous vector-valued functions on compact intervals can be constructed in the same way as are Riemann integrals of continuous complex-valued functions. Let P = { P k } ; = l E R", Q = {Qk};=lE R", and P 5 Q. Then, [ P , Q ] is a compact interval in R". For each k, partition the one-dimensional compact interval [Pk, Q k ]into mk subintervals with the nondecreasing set of endpoints P k = t k , 0 ? t k 3 ] ~ .t .k ., m, = Qk. Upon selecting a particular endpoint t k , ,r for each k , we get a point 1, = { t i ,

. . ., tn, pnl E [ P , Q1,

where p { p l , . . ., p,,} and 0 Ip <m LI { m l , .. . , mn}. Let Z, denote the n-dimensional subinterval [ t , - [ l , , t,], where [ l ] 5 p I m and [ l ] = (1, 1, . . . , l}. Set

We let 7c denote the collection of subintervals {Z,,}cl15,,sm tangular partition of [P, Q ] . Finally, we set A

(7c( =

and call

71

a rec-

max It, - r,-[lll. [IlspSm

Given two rectangular partitions 7c1 and 7c2 of [ P , Q ] , we say that 7c2 is a refjnenient of 7c1 if every subinterval in 7c1 is the union of a set of subintervals in 7 c 2 . Let f be a continuous function on [ P , Q ] into Y , where, as usual, [ P , QJ has the topology induced by R". Corresponding tofand any given rectangular partition of [ P , QJ,we can set up the Riemann sum

1.4. INTEGRATION

7

be a sequence of rectangular partitions of [P, Q] such Next, let that I n k ( -0. Then, { S ( f , n,)},",, is a Cauchy sequence in Y . Indeed, by Theorem 1.3-1, f is uniformly continuous on [P, Q]. Hence, given any y E r and E E R , , there exists an r] E R , such that

Here, nk,, is a partition of [P, Q] that is a refinement of both nk and n,, and g is a function on [P, Q] into Y such that, on each subinterval of nk,,, , g is a constant function of the form g(t) = f ( t , ) - f ( t e ) ,where t , and tC are fixed points satisfying I t , - t,l < r]. Hence, by (1) and (2),

s(f,

nk)

-

s(f,

= E d t , ) vO1

nm)

9

Ir

where the summation on p corresponds to the rectangular partition nk,,. Thus,

1 Y[g(tp)l c

y[s(f,nk) - s(f,

vO1

So, truly, {S(J nk)}p= is a Cauchy sequence in Y . Since Y is sequentially complete and separated, this sequence has a unique limit in Y , which we denote alternatively by

=I

Q

f(t) dt '[P. Q1

P

f(t) dt =

I

Q

fdl

P

and refer to as the (Riemann) integraloffon [ P , Q]. It is also a fact that the integral o f f is independent of the choice of the sequence {nk} of rectangular partitions. This can be shown by making use of the next lemma.

,

-

Lemma 1.4-1. Let {cpk}p= tend to 4 in Y and { $ k } p = tend to $ in v .Assume that, for ettery y E r, y(& - $ k ) 0 as k and m tend to infinity independently. Then 4 = $.

1. VECTOR-VALUED FUNCTIONS

8

If 4 # $, then by the separatedness of y(4 - $) > 0. But then,

PROOF.

such the

Y , there exists a y

E

r

Y ( 4 k - $m) = r ( + k - 4 + 4 - $ -k $ - $ m ) 2 y(4 - $1 - Y ( 4 k - 6)- r($ - $m), and the last two terms on the right-hand side tend to zero as k + 00 and m + 00. Hence lim Y ( + k - $ k ) 2 y(d, - $) > 0, which contradicts the hypothesis. 0 Now, let { n k } and {nk'} be two sequences of rectangular partitions of [ P , Q] such that I 7[k 1 + 0 and I 71', I + 0. By virtue of Lemma I .4- 1, to prove that S ( f , nk) and S ( f , nk') tend to the same limit, we need merely show that, for each y E r, y [ S ( f ,nk ) - S ( f , TC,')]-+O as k and m tend to infinity independently. But this can be done by setting S ( f , nk) - SCf, nm') = S ( g , nk, as in (2) and arguing exactly as before. We summarize the results obtained as far as follows.

Theorem 1.4-1. Let f be a continuousfunction that maps the compact interral [ P , Q] c R", where P 5 Q, into V . Then, for any sequence {nk} of rectangular partitions of [ P , Q] such that I nk 1 + 0,the Riemann sums S ( f , nk) tend in Y to a unique limit f ( t ) dt, called the integral o f f on [ P , Q]. Moreotler, the limit f ( t ) dt is independent ojthe choice of {q"k.

I$

When Q IP , we use the definition P

JpQ

f ( t ) d t '(-1)"s

f(t)dt. Q

We now list a number of facts, which are all easily established from the definition of ff f ( t ) dt. I n the following, both Y and W are separated sequentially complete locally convex spaces, and f and g are continuous functions mapping the compact interval [ P , Q] c R", where P 2 Q, into Y .

I . j$f ( t ) dt is the zero member of V if any component of P is equal to the corresponding component of Q. 11. Let F E [ Y ;W]. (For example, we may have F E Y ' = [ Y ;C].) Then, Q

f ( t ) dt

F P

1 Ff(t) d t . Q

=

P

111. For any continuous seminorm y on Y ,

I .5.

REPEATED INTEGRATION AND IMPROPER INTEGRALS

9

IV. Let Y = [ A ; El, where A and E are complex Banach spaces. Then, for any a E A ,

V. Let Y = [ H ; HI, where H is a complex Hilbert space with the inner product (., .). For any a, b E H ,

VI. Integration is a linear process. That is, for o[,

P E C,

VII. Let F be a continuous mapping of [ P , QJ into [ Y ; W ] .Let h denote the function t H F ( t ) f ( t ) , where t E [ P , Q ] . Then, for any sequence {7[k}km,1 of rectangular partitions of [ P , Q ] such that I l [ k I -,0, r F ( t ) f ( t ) dt A lim S ( h , nk) .JP

k-tm

exists as a limit in W and is independent of the choice of { 7 [ k } . VIII. Assume that {fj}y=lis a sequence of continuous Y-valued functions that converges uniformly on [ P , Q ] to the Y-valued functionf. (By this, we mean that, given any y E and E E R , , there exists an integer k such that y[f(t) - Act)]< E for all j > k and all t E [ P , Q ] . )Then,fis also continuous and

Problem 1.4-1. Prove the preceding eight assertions. 1.5. REPEATED INTEGRATION AND IMPROPER INTEGRALS

Under the assumptions of Theorem 1.4-1, the integral

can be written as the repeated integral

1.

10

VECTOR-VALUED FUNCTIONS

To see this, we first rewrite S(A n) as follows:

Set lnkl = max 1 SPkSmk

As

1 ~ " ) +O,

( k Pktk,Pr-I).

the innermost summation in (3) tends to

because, with all the components o f t fixed except for t , , t , H f ( f ) is a continuous mapping of [ P , , Q,] into Y . Now, (4)is a continuous function on [PI,Q,] x * . . x [Pn-l,Qn-l] into Y . Indeed, let At denote an increment in t such that t, remains unchanged. Then, by note IIi of the preceding section,

By the uniform continuity of A the right-hand side tends to zero as I At I + 0, which verifies our assertion. As a consequence, the two innermost summations in (3) tend to

J;;,'

(k-I

pw

dt,

as first Inn[+ O and then In"-'l -0.. Continuing in this way, we see that (3) tends to ( 2 ) as all the Ink/ are taken to zero one at a time in the order of decreasing k. To show that ( 1 ) is equal to ( 2 ) , choose y E r and E E R , arbitrarily. We may write

1.5.

REPEATED INTEGRATION AND IMPROPER INTEGRALS

11

Now, by what we have already shown in this section and by Theorem 1.4-1, we can choose a sufficiently fine rectangular partition n of [ P , Q] such that every term on the right-hand side is less than &/(n+ 1). [Indeed, for one such term, say, the pth term, we can choose a n p such that this term is less than &/(n+ 1) for every refinement of n p . Then, a n can be found that is a refinement of every np.] Hence, the left-hand side of (5) is less than &, which proves the equality between (1) and ( 2 ) . The order of integration in ( 2 ) can be changed in any fashion. This follows from the fact that we can change the order of summation in (3) and then take repeated limits as above to get another repeated integral equal to (1). The next theorem summarizes all this. Theorem 1.5-1. Let f be a continuous function that maps the compact interval [ P , Q] c R", where P I Q, into Y . Then,

IPQj(t) dt

=

1"'

dt,

. . IpQ"j(t) dt,.

PI

Moreover, the order of integration in the right-hand side can be changed in any fashion.

The rest of this section is devoted to a discussion of certain improper integrals. Let Q" E R," be such that one or more of its components are 00. We shall say that Q E R" tends to Q" and shall write Q + Q" if each component of Q tends to the corresponding component of Q". Now, let P E R" be such that P 5 Q" and assume that f is a Y-valued continuous function on [ P , Q ] for every Q E R" such that P I Q 5 Qm. Assume in addition that Q

lim j p f ( t ) d t

Q-Q"

(7)

exists in -Y and is independent of the fashion in which Q -+ Q". [Thus, for example, the components of Q could tend to those of Q" simultaneously or one at a time and in any order without changing (7).] The improper integral

is defined to be the limit (7). In quite the same way, we can define the improper integral (9)

12

1.

VECTOR-VALUED FUNCTIONS

where P-" is an n-tuple with - 00 for some or all of its components and real numbers for the remaining components. When all the components of P-" are -a and those of Qm are co, we write Q"

lp--f ( t ) dt

=

I

f ( t ) dt =

R"

I

fdt.

(10)

R"

It is readily seen that all the assertions of notes I-VI of Section 1.4 continue to hold for (8)-(10). Now, however, the right-hand side of the inequality in note I11 may become co.

Problem 1.5-1. Prove the two statements of the preceding paragraph.

1.6. DIFFERENTIATION

Let Atk E R . In the following, At I k will denote a member of R" all of whose components are zero except for the kth, which is instead A t k . Thus, for any t = {tl, . . ., t,} E R", t -k Atlk = { t i , . . . , t k - 1 ,

fk

-k Atk,

tk+,,

. . ., t,}.

Let f be a Y-valued function on some subset of R". (We remind the reader that throughout this chapter, we are assuming that Y is a separated sequentially complete locally convex space with r as a generating family of seminorms for the topology of Y . ) We say thatfis diflerentiable at the point t with respect to fk iffis defined on some neighborhood o f t and if the quantity f ( t -k At I k ) - f(r) Atk

converges to a limit in Y as I Atkl + 0. That limit is called the deriaative of f w i t h respect to t k evaluated at t and is denoted by a k f ( t )as well as by a , , f ( t ) . Let R be a subset of R".f is said to be differentiable on R with respect to t, iffis differentiable at every t E R; in this case, its derivative a k f = a,,fis also a Y-valued function on R. If a k f is in turn differentiable on R with respect to, say, t i , the second derivative aj akf of f is defined as the derivative of akf on R with respect to t i . Continuing in this way, we obtain the higher derivatives off. Integration and differentiation can be related as follows.

{Qk}z=l

Theorem 1.6-1. Let P = {Pk}i=1E R" and Q = E R" be such that Pk = Qk f o r all components except the j t h component, for which we have

1.6. DIFFERENTIATION

13

instead P j < Qj. Let f be a "Y-valued function having a continuous derivative 0jf on [P, Q]. Then, (1) PROOF. The left-hand side of (I) exists because of the continuity of 0J(t) as a function of t l : Now, let "Y' be the dual of"Y and let FE "Y' be arbitrary. It follows from the definition of oJ that F ojf(t) = OJ Ff(t). So, by note II of Section 1.4 and the fundamental theorem of integral calculus,

But, the weak topology of "Y separates "Y (Appendix D7), and thus (I) is obtained. <> In regard to a change in the order of differentiation in a second derivative, we have the following result. Theorem 1.6-2. Let f be a "Y-z'alued function on a subset of R 2. If oI!, 02f, and 02 0tf exist and are continuous on a neighborhood n ofa fixed point t E R 2, then 0t 02f exists at t and

(2) Note. We could apply an argument such as that of the preceding proof. But this would only establish that 0t 02f(t) exists as a limit in the weak topology of "Y. To conclude that 01 ol!(t) exists in the initial topology of "Y, we proceed as follows. PROOF. Set t = {xo, Yo}. In the following, hER and IE R are so restricted that {x o + h, Yo + I} remains in a compact interval A contained in nand containing {x o, Yo} in its interior A. We want to show that

(3)

exists and has the value 02 oI!(xo, Yo). Now, (3) may be written as lim lim (ljhl)[f(xo h-Ol-O

+ h, Yo + I) -

f(xo

+ h, Yo) -

f(xo, Yo

+ I) + f(xo, Yo)].

14

1. VECTOR-VALUED FUNCTIONS

We may invoke Theorem 1.6-1 to rewrite this expression as

+ z , yo + I ) - d,f(xo + z , y o ) ] d z

lim lim ( l l h l ) f[dlf(xo h-0

1-0

0 h

=

lim lim ( l l h l ) h-0

But, for any y

E

5 SUP

14 2 I:

140

I I

dz 0

d2 d,f(xo

0

+ z , yo + i)d i .

r, we may employ note 111 of Section

Y P z dlf(X0 +

2 9

Yo

+ C) - 4d,f(xo

7

1.4 to obtain

Yo)].

I n view of the continuity of a, a,f at {xo, y o } , the last expression tends to zero as first I and then h tend to zero. This proves that (3) exists as a limit in Y and is equal to d 2 d l f ( x o ,yo). 0

Let k = { k l , . . . , k,} be a nonnegative integer in R". Dk denotes the differential operator Dk 4 8;: . . . a;.. (4) We also use the notation Dkf=,f(k)as well as Dkf(t) = D,"f(t) = f ' k ' ( t ) . Throughout this book, we adhere to the following conventions. Whenever k is used as in (4), the notation I k I will denote the sum k , + . . . + k,, rather than the magnitude of k as a member of R". Moreover, we shall refer to k as the order of Dk. (This is in contrast to the usual practice of referring to I k 1 as the order of Dk.) Whenever we say that the derivatives of a functionfup to the order k are continuous on a set R, we mean that the functionsf, akf, aj akJ . . . , DkA obtained by successively applying to f the various operators ak to obtain 0% exist and are continuous on R. If R is an open set, it follows from Theorem 1.6-2 that any change in the order of differentiation in Dkfwill not alter Dkfon R. The same is true when R is a compact interval, because the values of Dhf on the boundary on R are continuous extensions of its values on the interior of R. We shall say that a Y-valued function defined on a subset of R" is smooth if the function possesses continuous derivatives of all orders at all points of its domain.

1.6.

15

DIFFERENTIATION

Theorem 1.6-3. Let W be a separated locally conuex space. Also, let F and

f be, respectioely, [Y ; W]-valuedand Y-valuedfunctions having the continuous derivatives ak F and akfon a neighborhood of the point t E R". Then, at t, ak(Ff) =

flf +

(5)

akJ

[Here, Ff denotes the function X H F(x)f (x).] Moreover, f F and f have continuous derivatives up to order k on a neighborhood of apoint t , then, at t, we have

where

Note. The last quantity is the n-dimensional binomial coeficient. For n = 1, we have

where m and q are nonnegative integers in R with q < m. Also, ( 6 ) is called Leibniz's rulefor the differentiation of aproduct. PROOF. In this proof, Atk tends t o zero inside an interval so small that Atk HF(t + At I k ) and Atk H f ( t + At I k ) are continuous and therefore bounded functions on that interval (Theorem 1.3-1). The left-hand side of (5) is the limit of W (if it exists) of

Since [ Y ;W ]possesses the bounded topology, the assumption that F has a derivative at t means that

tends in [ Y ;W ]to G(0) the assumption on f ,

dk

g(Atk)

F(t) uniformly on the bounded sets in Y .By

f( t + At 1 k ) ,

Ark # 0

tends in Y to g(O) 4 f(r). Moreover, {g(Atk)},where Atk traverses its permissible values, is a bounded set in Y . Now, G(Afk).dArk)

- G(o).do) =z [G(Atk)- G ( o ) b ( A t k )

+ G(0)[dArk)- do)].

1.

16

VECTOR-VALUED FUNCTIONS

Thus, both terms on the right-hand side tend to 0 in W , which shows that the first term in (7) tends to (a, F)(t)f(t). The second term in (7) clearly tends to F ( t ) f(t) in W .Thus, ( 5 ) is established. Equation (6) is established by repeated application of (5). 0

a,

As a last consideration, we take up differentiation under the integral sign.

Theorem 1.6-4. Let [ P , Q] be a compact interval in R" and E an open set in R ; t and x will denote variables in [P, Q] and E,respectively. Assume that f is a continuous Y-raluedfuncrion on [P, Q] x E and that a, fexists and is continuous on [P, Q] x E.Set .Q

g ( x ) = J f( t , X) dt,

-

x E .;

(8)

P

Then, a,g exists at each point of E and

PROOF.Fix X E E and restrict A X E R so that x consider

+ A X E S . For

Ax # 0,

By Theorem 1.3-1, d,fis uniformly continuous on [P, Q] x E.Therefore, given any 8 > 0 and any y E r, there exists an q E R + such that

for every t E [P, Q] and all 5 such that 15 - X I < 4. Appealing to note 111 of Section 1.4 and using (lo), we may write

I dt 0

y[hAx(x)l

5

P

1

IlI+AxY[a,f(t, r)

- axf(t.

dr

<&

for all IAxl < q. This proves the theorem. 0 Problem 1.6-1. Let W , as well as Y , be a separated sequentially complete locally convex space. Also, let F and f be, respectively, [ V ;W]-valued and

1.7.

BANACH-SPACE-VALUED ANALYTIC FUNCTIONS

17

$'--valued functions such that F , L dk F, and dkf are continuous on a compact interval [ P , Q] c R". Establish the formula for integration byparts:

Next, assume that F andfare smooth on R" and that at least one of them has a compact support. Show that, for any nonnegative integer k E R",

( D k F ) f d t = (- 1)lk1

I FDvddt. R"

Problem 1.6-2. Let [ P , Q] and E be as in Theorem 1.6-4 and let f be a continuous Y-valued function on [ P , Q] x Z. Define g by (8). Show that g is a continuous function on E. Problem 1.6-3. Let {f,},"= be a sequence of continuous Y-valued functions on an open set R c R". With dk denoting a differentiation, assume that dkf, exists and is continuous on R for every m. Also, assume that {f,} converges uniformly on R to a functi0n.A whereas {d,f,} converges uniformly on R to a functiong. Then, d,fexists on R, and a,f= g. 1.7. BANACH-SPACE-VALUED ANALYTIC FUNCTIONS

We turn to a discussion of analytic functions that take their values in a complex Banach space A . Let R be an open set in C and letfbe a mapping of R into A . The mapping f is said to be diferentiable (or weaklydiferentiable) at a given point ( E R if, as A( + 0 in C ,

tends to a limit Of(()E A in the topology (or respectively the weak topology) of A and if this limit is independent of the path along which A( tends to zero. Of(()is called the deriratitje (or respectively the weak deriratire) o f . f a t (. Alternative notations that we use for Of(()are D , f ( ( ) and f(')((). The differentiability o f f a t ( immediately implies its continuity at 4'. Iff possesses a derivative (or weak derivative) at every point of R, then f is said to be analytic (or weakly analytic) on R. At times, we say that,f is analytic at a point [; by this, we mean that f i s analytic at every point of some open neighborhood of (. If B is another complex Banach space and if F i s an [ A ; B]-valued function on R, then, as A( +O, F ( i + A 0 - F(C) (2)

1. VECTOR-VALUED FUNCTIONS

18

may converge to a limit D F ( 0 4 DrF(() F"'(() in [ A ;B] with respect to any one of three topologies in [ A ;B], namely the uniform operator topology, the strong operator topology, and the weak operator topology (see Appendix D1 I), in such a way that the limit is independent of the path along which AC tends to zero. In this case, we say that F possesses respectively either a derivative a strong derivative, or a weak derivative at [. When this is so at every point of R, we say that F is respectively either analytic, strongly analytic, or weakly analytic on R. Clearly, the analyticity off on R implies its weak analyticity, whereas the analyticity of F on R implies its strong analyticity, which in turn implies its weak analyticity. I t is quite a useful fact that the reverse implications are also true, as is stated later, by Theorem 1.7-1. To show this, we first prove the following. Lemma 1.7-1. Let g be a complex-z.alued analytic function on an open set R c C and let Z be a closed disc {( E C : 1 ( - lo I 5 r, lo E C, r > 0 } contained in R. Then, there exists a j n i t e number M = M ( f , E) depending on f and Z such that, for euery (, + LY, and ( + P in the interior k of Z, we have 1 Q(i, a, PJI 5 M , where

PROOF.Let S denote the boundary of 2 . We can choose a circle P in R which encircles E such that

{ I T - (1 : 7 E P , [ E S E I } > 0. Then, upon fixing (, ( + a, and 4' + P as points in 2, we may use Cauchy's d4

LJ

integral formula to write

Theorem 1.7-1. An A-i.alued function on an open set R c C is analytic on R ifand only i f i t is weakly analytic on R. Similarly, an [ A ;B]-ralued,functionis analytic on R ifand only f i t is strongly analytic on R, and this is the case ifand only ifit is weakly analytic on R.

PROOF.We prove only the second sentence, the proof of the first one being quite similar. In view of our previous remarks, we need merely show that the

1.7.

BANACH-SPACE-VALUED ANALYTIC FUNCTIONS

19

weak analyticity of the [ A ; B]-valued function F on R implies its analyticity on Q. Let a E A and b' E B', where B' is the dual of B. Set

Choose Z as in Lemma 1.7-1 and let i,5 + a, and 5 + P be points in 8. By Lemma 1.7-1, there exists a constant M depending on a, b', F, and E but independent of (', a, and P such that

I b"C, By Appendix D12,

a, P)a

llR(c, a,

I

M.

P>aIIB

where M , depends on a, F, and E but is independent of 5, a, and P. An application of the principle of uniform boundedness (Appendix D12) now shows that

llR(5?

P>Il[A, B ]

M2

I

where M , depends o n F a n d E but is also independent of 5,a, and P. Now, let E E R , be arbitrarily chosen. For all permissible a and B such that I a I and I /3 I are both less than & / 2 M , ,we have that I a - P I < 4 M 2 , so that

By virtue of the completeness of [ A ; B], this implies that, as a + O ,

[m+ 4 - F(0ll. tends to a limit in the uniform operator topology of [A; B]. Hence, F has a derivative at C. Since ican be selected as any point of R by appropriately choosing Z, F is analytic on R. 0 As a consequence of Theorem 1.7-1, many of the results for complexvalued analytic functions can be carried directly over to A-valued or [ A ; B]valued analytic functions. For example, let F be an [A; B]-valued analytic function on an open set R and let a E A and b' E B'. Then, b'F(.)a is a complexvalued analytic function on R, so that Dkb'F([)a exists for every k and every ( E R. This implies that D k Fis weakly analytic and therefore analytic on R for every k, which shows that F i s smooth on R. We finally observe that an adaptation of the proof of Theorem 1.6-3 establishes Liebniz's rule [see (6) of Section 1.61 for the differentiation of Ff, where F is an [ A ;B]-valued analytic function and f is an A-valued analytic function.

1.

20

VECTOR-VALUED FUNCTIONS

1.8. CONTOUR INTEGRATION

Theorem 1.7-1 also allows us to extend contour integration to Banachspace-valued functions. We do so in this section for the integral of an Avalued function f o n a contour P in C. Since [A; B ] is also a Banach space, our results immediately extend to an [ A ; B]-valued function F on P. The integral j p d[ of a continuous A-valued function f on a contour P in C is defined and shown to exist exactly as in the case of a complex-valued function (Copson, 1962, pp. 52-59). We also have, as in the scalar case, the estimate

f(c)

As an immediate consequence of the definition of J p f ( ( )dc as a limit of certain Riemann sums, we can state that, for any a E A, a’ E A’, and b’ E B’,

and

Contour integration is a linear process as in note VI of Section 1.4. Moreover,

where -P is the contour obtained by reversing the orientation of P.

Theorem 1.8-1 (Cauchy’s theorem). Let R be a simply connected open set in C, and let P be a closed contour in R. I f f is analytic on R, then

PROOF.When P is a closed contour in $2,the right-hand side of (2) is zero according to Cauchy’s theorem for complex-valued functions. Equation (4) now follows from the fact that the weak topology of A separates A (Appendix D7). 0 Arguments like that of the preceding proof can be used to extend other standard results from the theory of complex-valued analytic functions.

I .8.

CONTOUR INTEGRATION

21

Theorem 1.8-2 (Cauchy's integral formula). Let R, P, and f be as in Theorem 1.8-1 and let ( E C be apoint inside P. Then,for each nonnegative integer k E R ,

Theorem 1.8-3. Let {f,},"= be a sequence of A-ralued analyticfunctions on an open set R c C such that f, + f i n A uniformly on each compact subset of R. Then, f is also an A-valued analytic function on R and for each nonnegative integer k fik) +f ( k ) in A uniformly on each compact subset of R.

[For this theorem, an argument like the proof of Theorem 1.8-1 would only prove that fik'(() + f (k)(lJ in the weak topology of A . However, an estimation of f,"')(()- f ( k ) ( ( ) using (5) leads to our stronger conclusion.]

{c,

Theorem 1.8-4. Let ?}I+ f([,T) be a continuous A-valued function on R x P, where 4' E R, r E P, R is an open set in C , and P is a contour in C. Assume thatf r ) is an analyticfunction on SZ for each r E P. Set (a,

G(C)

1f ( T , 4 dr. P

Then, G is an A-valued analytic function on R, and Gtk)([)= P

k = I , 2, . . . .

D,"f([, r) dr,

In the next theorem, P is an oriented path in C extending to infinity such that any finite portion P,,obtained by tracing P from one of its points to another is a contour. Let {P,,},"=, be a collection of such finite portions of P with the properties that P,,c P,,,l and Pn= P. We define

u.

I p f ( r ) dr 4 lim n-cc

P,

f(r) dr

if the limit exists. Upon combining Theorems 1.8-3 and 1.8-4, we obtain the following.

Theorem 1.8-5. Assume that f satisfies the hypothesis of Theorem 1.8-4 on R x Pnfor each n. Suppose that, as n + co, J P n f ((, r) dr conoerges uniformly with respect to all 5 in each compact subset of R. Define G by (6). Then, the conclusion of Theorem 1.8-4 holds once again.

Problem 1.8-1. Prove Theorems 1.8-2-1.8-4.

22

1. VECTOR-VALUED FUNCTIONS

Problem 1.8-2. Let P be a contour starting at w E C and ending at z E C. Assume that the [ A ; B]-valued function F and the A-valued function f are analytic on P. Show that

and

Chapter 2

Integration with Vector-Valued Functions and Operator-Valued Measures

2.1. INTRODUCTION

An essential tool needed in our subsequent development of an admittance formulism for time-invariant passive systems is a certain theory of integration due to Hackenbroch (1968), wherein functions taking their values in a given Banach space A are integrated with respect to measures that take their values in a space of operators mapping A into another Banach space B. Chapter 2 is devoted to a presentation of this theory. 2.2. OPERATOR-VALUED MEASURES

Throughout this chapter, T is an arbitrary nonvoid set, 6 is a a-algebra of subsets of T, n = {Ek};=is an arbitrary partition of T, and 1is the collection of all partitions n. (See Appendix G , Sections, GI, G2, and G7.) As 23

24

2.

INTEGRATION WITH VECTOR MEASURES

always, A and B are complex Banach spaces, and H is a complex Hilbert space with the inner product -). The concept of the total variation l p l ( T ) = Varp of p on T, where p is a complex measure, can be extended to any mapping P of 6 into [A; B]. ( a ,

Total variation of P on T !& Var P

Similarly, the semivariation is defined as follows: Semivariation of P on T

War P

These concepts are commonly used in the theory of operator-valued measures (Dinculeanu, 1967). On the other hand, Hackenbroch bases his theory of integration on still another concept of variation, namely, the following: Scalar semivariation of P on T

This definition of SSVar P continues to have a sense when P is replaced by a mapping Non 6 that takes its values in an arbitrary Banach space A, and not merely in the space [A ; B] of operators. The integration of certain complex-valued functions with respect to an A-valued function N on 6 or an [A; B]-valued function P on 6 can be defined as an extension of the integration of the so-called simple functions so long as N and P are additive and have finite scalar semivariations. The process is similar to the usual construction of the Riemann integral of a continuous function. Let us be explicit. As is ;Ind;\tik..in Append;\x G8, a simpre functian f fram T into A is any mapping of the formf= x k a k ,yEr, where a k E A, {Ek)E 9, and zEkdenotes the characteristic function of Ek. We let %,(A) a B,(T, 6 ;A) denote the linear space of all simple functions from T into A. Moreover, G(A) is taken to be the Banach space of all bounded A-valued functionsffrom T into A with the norm I1 * IIG(A) where IlfllG(A)

A I E T Ilf(t)llA*

2.2.

25

OPERATOR-VALUED MEASURES

Hence, Bo(A) c G ( A ) . Finally, % ( A ) will denote the closure of g0(A)in G ( A ) . When A = C , we simplify our notation by setting Yo(C) = 9,,B(C) = 9,and G ( C ) = G. % ( A )is in general a proper subspace of @ A ) . A mapping N of (5 into A is said to be additive if, for any pair of disjoint sets E, F E (5, we have N ( E u F ) = N ( E ) + N ( F ) . In this case, we can define the integral of any complex-valued simple function f = ark xEk, where ak E C , with respect to N by means of the expression

IT

f(?)

dNt

1

N(Ek)ak

a

k

The subscript T on the integral sign is at times dropped when the set T on which the integration is taking place is evident. The subscript t on N signifies that the points in T a r e denoted by t . Here are some immediate results of our definitions. The function

f(0

f+N,

(1)

is a linear mapping of gointo A , and, for all f E Yo,

I sup If(t)I SSVar N . A

t s T

Now, assume that SSVar N < CQ. In accordance with Appendix D, Sections D2 and D5,this implies that the mapping ( I ) has a unique extension that is a continuous linear mapping of B into A . We use the same symbolism for the extended mapping. The inequality (2) continues to hold for all f E 9.Assume still further that N is the mapping P of C into [ A ; B]. For any U E A and b‘ E B’, where B‘ is the dual of B, and for all f E 9,we have

(3) and

[

b‘ jdP‘

4

u = Sd(b’P, 0)

Here, P( .)a is a B-valued additive function on

(5,

f(0.

(4)

and

SSVar P( .)a 5 (lull SSVar P. Also, b’P(.)ais a complex-valued additive function on

(5,

and

SSVar b‘P( .)a 5 11b’II llall SSVar P.

In regard to the functions on (5, our attention will be primarily confined to those functions that take their values in an operator space [ A ; B] and possess the following a-additivity property.

2.

26

INTEGRATION WITH VECTOR MEASURES

Definition 2.2-1. A function P on (5 into [ A ; B ] is said to be a-additive in the strong operator topology of [ A ; B] if it is additive and, given any sequence {Ek}2=lc (5 such that Ek n E j is the void set whenever k Zj,we have that

.( kgl

m

c

= k=

Ek)a

1 P(Ek)a

for every a E A . When this is the case, P is called an operator-valued measure or more explicitly an [ A ; B]-valued measure. If, in addition, A = B = H and the range of P is contained in the space [ H ; H I , of positive operators (see Appendix D I5), P is called a PO measure or apositive-operator-valuedmeasure. Note that P is an [ A ; B]-valued measure if and only if P is additive as a function on (5 into [ A ; B ] and, for every increasing sequence {Ek}F=l in 0 (i.e., Ek c Ek+ for all k ) and for all a E A , P

u Ek a

=

( I

lim P(Ek)a. k-tm

We now investigate in some detail the properties of operator-valued measures and PO measures. Let {Ek}p=,be an increasing sequence in 0 with UkEk = T . Then, for any PO measure P , iiP(Ek)Il

=

sup

(P(Ek)a,0 )

5 sup ( P ( T ) a ,a ) = IIp(T>II* llall = 1

1 1 4= 1

(5)

Moreover, since (P(Ek)a,a) + ( P (T ) a ,a ) for each a E H , it follows that IIp(Ek)II

+

IIp(T)II,

k

+

00.

(6)

Lemma 2.2-1. If M E [ H ; H I + and Mk E [ H ; H I , , where k = 1, 2, . . ., and $ as k + 00, ( M k a ,a) increases monotonically to the limit ( M a , a ) for every a E H , then Mk + M in the strong operator topology of [ H ; HI.

PROOF.If Q E [ H ; H I , and a E H , we may employ Schwarz’s inequality (Appendix A6) to write

IIQa1I4 =

\(eatQaII

I

(pa,a>(QQa,Qa).

But ( Q Q a , Q45 IlQII IIQaII2, and so

IIQal12 5 (Qa,a)IlQll. (7) We may replace Q by M - M kbecause M - h f k E [ H ; H I , according to the hypothesis. Moreover, ( ( M - Mk)a, a) -+ 0 as k + 00. On the other hand, upon invoking the polarization equation (Appendix A7) and making two applications of the principle of uniform boundedness (Appendix D12), we see that IIM - Mkll is bounded for all k . This proves the lemma. 0

2.2.

27

OPERATOR-VALUED MEASURES

Theorem 2.2-1. Let P be an additive mapping of follo wing three assertions are equivalent.

(5

into [ H ; HI,.

The

(i) P is a P O measure on (5. (ii) For all a, b E H , (P( .)a, b) is a complex measure on (5. (iii) For every a E H , ( P ( ' ) a ,a) is a positivejnite measure on (5. Note. As is indicated in Appendix G4, complex and positive measures are by definition a-additive.

PROOF.That (i) implies (ii) and (ii) implies (iii) is clear. To show that (iii) implies (i), we need merely prove that P is a-additive in the strong operator topology of [ H ; H I . Let {&}?=1 be an increasing sequence in (5, and set E = U k E k . Hence, E E (5. Consequently, P(Ek) E [ H ; H I , and P ( E ) E [ H ; H I , . Also, (P(E,)a, a ) increases monotonically to (P(E)a,a) because of the positivity and a-additivity of the measure (P( .)a,a). Hence, by Lemma 2.2-1, P ( E k ) P ( E ) in the strong operator topology of [ H ;HI. 0 --f

Theorem 2.2-2. If P : (5:

-

[ A ; B] is an operator-valued measure, then

SSVar P I 4 sup IIP(E)IIIA;Bl< 00.

(8)

EEO

PROOF.Let F be an arbitrary member of the dual of [ A ; B]. Then, FP 4 FP( .) is a complex measure on (5. Let FP = L + iM,where L is the real part of FP and M the imaginary part. For any 7c = {Ek};= E 1,we may write

1I

L(Ek)

k

I

=

1

+

L(Ek)

-

=L(u+Ek)

1'

-

u' (1- u-)

1

- L(Ek)

L(u

-Ek)?

where and and are taken over those k for which L(Ek)2 0 [respectively L(E,) < 01. Therefore, R E

9

1I

L(Ek)

1

= RE

2

u

[L(

+Ek)

- L(

u-

Ek)]

I 2 s u p IL(E)I. EEO

A similar inequality holds for M , and therefore SSVar F P s Var F P Isup

1 IL(Ek)I + sup 1 I M(&)I

R E 9

R E

9

S 2 s u p IL(E)I + 2 s u p IM(E)I EEO

EEO

I4 sup 1 F P ( E ) I EEO

I4

IlFll SUP IIP(E)II. EEO

2.

28

INTEGRATION WITH VECTOR MEASURES

sup SSVar FP I 4 sup IIP(E)II.

IlFll = 1

EEC

But SUD

SSVar FP

According to Appendix D 13, IlFll = 1

I 1

I=1 1

P(Ek)c(k

P(Ek)ak

1

[ A : B]r

and therefore the right-hand side of (9) is equal to SSVar P.This establishes the first inequality of (8). To obtain the second inequality, let b ' E B' and U E A. According to Appendix G7, the complex measure b'Pa = b'P( * ) asatisfies sup I b'P(E)al I Var b'Pa < a. EE&

Two applications of the principle of uniform boundedness (Appendix D12) complete the proof. 0 Theorem 2.2-3. U P :0: -+ [ H ; HI, is a PO measure, then SSVar P

=

IIP(T)IIIH:Hl.

(10)

PROOF. Choose arbitrarily a partition 7t = {Ek};= E 1,two members a and b in H , and the complex numbers c(k such that I c(kl I1, where k = 1, . .., r. We may write

1(

k= 1

cckP(Ek)a,

b,

1

1

I(P(Ek)a,

b)l.

(11)

By the Schwarz inequality (Appendix A6), the right-hand side is bounded by (P(Ek)a,a)'"(P(Ek)b, b)1/2. But, by the Schwarz inequality for sums, the last expression is bounded by

[c(P(E&?,

(P(&)b, b)]1'2= (P(T)U,a)'I2(P(T)b, b)'"

I llP(T)Il llall llbll. Take a supremum of the left-hand side of (1 1) over all 71 E 9, all clk such that (akl 5 1, and all a, b E H such that ((a(( = I(b(l= 1. This yields SSVar P IIIP(T))I.On the other hand, upon choosing r = I , a1 = 1, and El = Tin the definition of SSVar P , we see that SSVar P 2 IIP(T)II. 0

2.3.

29

6-FINITE OPERATOR-VALUED MEASURES

Problem 2.2-1. A fact we will use later on is the following. Let T = R" and let 0. be the collection of Bore1 subsets of T. For q = {qk};,, E R" and t = { t k } ; = E R", we set qt = x k qk t k . Then, for fixed q, the function tweirlr is a member of 8. Show this. Problem 2.2-2. Let P be a mapping of 0. into [ A ; B]. Show that IIP(T)II I SSVar P I SVar P 5 Var P I co. Problem 2.2-3. With P being an additive mapping of (I into [ A ; B], assume that SVar P < co. Show that an integral J dP,f(t) can be defined for any f E % ( A ) as an extension of the integrals (defined in a natural way) of the functions in %,,(A). Also, show that

and, for any b' E B', b'

1

dP,f(t) = d(b'P,)f(t),

where b'P(.) is an A'-valued additive function on 0. and SVar b'P( .) I Ilb'IIB.SVar P.

[Thus, upon assuming that SVar P < co,we obtain hereby a means of integrating A-valued functions with respect to [ A ; B]-valued measures. However, Hackenbroch's theory encompasses greater generality so far as the measures P are concerned; it takes into account certain measures P for which SSVar P < 03 but SVar P = co. For an example of such a measure, see Hackenbroch (1968, pp. 332-333.)1

2.3. U-FINITE OPERATOR-VALUED MEASURES

Let { Tk}km, be an increasing sequence of sets with TkE (I and Set 0.k

'

{ E E 0.: E

C

Tk},

k

and

It follows that (I, is a a-algebra of subsets in T k .

= 1,

2,

a .

U k

Tk = T.

30

2.

INTEGRATION WITH VECTOR MEASURES

Definition 2.3-1. Let P be a mapping of 0, into [ H : H I , such that the restriction of P to each 0, is a PO measure on 0,. Then, P is called a a-finite PO measure on 0,. As an example of a a-finite PO measure, we may take T = R", Tk= { t E R": I t 1 5 k } , 6,equal to the collection of Borel subsets of Tk,and P equal to the Lebesgue measure on 0 , . P is finite on each bounded Borel subset of R" but is not defined on all the Borel subsets of R".

Throughout this section, P will always be a a-finite PO measure, and h will be a member of 9. It follows that lh(.)l E 93 also. By virtue of Theorem 2.2-3,

exists as a member of [ H ; HI for each k in accordance with the preceding section. Definition 2.3-2. h is said to be integrable with respect to P if, as k

+ CO,

( 1 ) converges in the strong operator topology of [ H ;H I ; the limit is denoted

by

(2) This defines a linear mapping h H J d P , h(t) of Y into [ H ; H I . Upon invoking (4) of the preceding section, we get, for all a, b E H ,

(1

d P , h(t)a, b )

= (lim k - a , JT,

= lim

d(P,a, b)h(t)

JT

k-rm

dP, h(t)u, b )

L

L? j d ( P , a , b)h(t).

(3)

Here, ( P ( .)a, b ) is an additive complex-valued function on 0 , whose restriction to each 0, is a complex measure. Similarly,

Theorem 2.3-1. Let g be a nonnegative.function in 9. g is integrable with respect to P if and only if the sequence { Fk}p=I , where Fk 4

I

dP,g(t) E W; HI,

Tk

is bounded in the strong operator topology.

2.3.

Q-FINITE OPERATOR-VALUED MEASURES

31

Note. By the principle on uniform boundedness, the boundedness of {Fk} in the strong operator topology is equivalent to the boundedness of {Fk}in the uniform operator topology.

PROOF.Since convergent sequences are bounded, the " only if" part of the theorem is clear. Conversely, assume that {Fk} is bounded in the strong operator topology. Clearly, (Fka,a ) 2 0 for every a E H , and therefore FkE [ H ; H I , . Similarly, F,,, - Fk E [ H ; H I , whenever m k. Also, by the principle of uniform boundedness again, llFkll s M , where M is a constant not depending on k . Moreover, upon appealing to (7) of the preceding section, we may write, for each a E H ,

=-

II(Fm

- Fk)a112

5 2 M ( ( F m - Fkb, a)*

The right-hand side tends to zero as k -, co because ( F k a ,a) is an increasing bounded sequence. Therefore, {Fk)is a Cauchy sequence in the strong operator topology and hence must converge because of the sequential completeness of [ H , H ] under that topology (Appendix D l l ) . 0 Theorem 2.3-2. If g E $4 is a nonnegative function and is integrable with respect to P and if h E $4 is such that I h( t ) I _< M g ( t )f o r all t E T, where M is a constant, then h is integrable with respect to P. Moreover,

PROOF.We may decompose h into h = h1

- h, + ih, - ih4,

(6)

where, for each j = 1, 2, 3, 4, hi is a nonnegative function in 3 ' and h,(t) M g ( t ) . Hence, for each k and j ,

(See Appendix 015). This inequality coupled with Theorem 2.3-1 shows that the hi,and therefore h as well, are integrable with respect to P. The estimate (5) follows from (7) and Appendix D l l . 0 The last theorem implies that h is integrable with respect to P whenever lh(*)I is.

32

2. INTEGRA nON WITH VECTOR MEASURES

Theorem 2.3-3. Let g E r§ be a nonnegative function that is integrable with respect to P. Define a mapping Q on (t as follows:

~

Q(E)

JdPtg(t) ~ f dPtxit)g(t), E

EE

(8)

(t,

T

where XE is the characteristic function for E. Then, Q is a PO measure on (t. PROOF. Observe that XEg satisfies the hypothesis of Theorem 2.3-2, so that the right-hand side of (8) exists as a member of [H; H). In fact, Q(E) E [H; H)+ , since

(Q(E)a, a) =

t «r,«.

a) XE(t)g(t) ;;::: 0

for every a E H. It is straightforward to show that Q is additive on (t. Next, we shall show that, for any a E H, (Q(. )a, a) is a positive finite measure on (t. To do this, we have only to show that (Q( . )a, a) is e-additive (Appendix G4). For any J E (t, XJ g is integrable with respect to P according to Theorem 2.3-2. Therefore, (Q(J)a, a) = lim .

m-+oo

f .ur,«, a)XJ(t)g(t),

Now, let {Ed be an increasing sequence in also. Note that

f

Tm

J

E (t.

(9)

Tm

(t

with Uk E k = E. Hence, E

E (t

.ur,« a) XE.(t)g(t)

is an increasing function of both m and k. By virtue of (9), given any s > 0, we can choose an m such that

o~

f d(Pta, a)xit)g(t) - f T

d(Pta, a)XE(t)g(t) <

Tm

teo

Then, we can choose a k such that

os

fr.; d(Pta, a)XE(t)g(t) - fr., .ur,«, a)XE.(t)g(t)

~ (P(Tm

11

E\Ek)a, a)sup Ig(t)1 <

te

teT

because of the e-additivity of (P( . )a, a) on (tm' Hence,

o~ s

(Q(E)a, a) - (Q(Ek)a, a)

f «r,«, a)XE(t)g(t) - fr.; d(Pta, a)XE.(t)g(t) < e. T

2.3. a-FINITE OPERATOR VALUED MEASURES

33

So, truly, (Q( .)a, a) is a positive finite measure on 6. By virtue of Theorem 2.2-1, Q is a PO measure on 6. 0 The preceding theorem implies that the integrability of h with respect to P and the value of J dP, h(t) do not depend on the choice of the sequence { Tk} . Indeed, let {Ek}km, be another increasing sequence in Emsuch that Ek = T. Let g E 9 be nonnegative and integrable with respect to P . Then, by the a-additivity of the PO measure Q defined by (8), as k --f co, Q(Ek)-, Q(T)or equivalently

uk

in the strong operator topology. Upon decomposing h in accordance with (6), we see that the same is true when g is replaced by h.

Theorem 2.3-4. Let P, g , and h be as in Theorem 2.3-2. Let Q be the PO measure defned by (8). Set E, = { t E T :g ( t ) # 0). Then,

where x is the characteristic function for E, and it is understood that x(t)h(t)/ g ( t ) = 0 for t # E , . E,

PROOF.We have from the theory of measurable real-valued functions that E C.Moreover, it is straightforward to show that the condition

I X ( t ) h ( t ) / d t )I 5

M 9

(1 1)

where M is a constant, implies that Xh/g E 9. [See Rudin ( I 966, pp. 8-1 6 ) . ] Now, the left-hand side of (10) exists by virtue of Theorem 2.3-2. In view of (1 1) and the fact that any constant function is integrable with respect to Q, we see that the right-hand side of (10) also exists by Theorem 2.3-2 again. Upon appealing to (3), we may write, for any a, b E H ,

By a standard result for scalar integrals (Dunford and Schwartz, 1966, p. 180) the right-hand side of (12) is equal to

where, for any E E 6,

2.

34

INTEGRATION WITH VECTOR MEASURES

Upon combining (12)-( 14) and noting that a and b are arbitrary, we obtain the equality in (lo). 0 Problem 2.3-1. Verify the first two sentences in the proof of Theorem 2.3-4.

2.4. TENSOR PRODUCTS AND VECTOR-VALUED FUNCTIONS

The objective of this section is to relate the tensor product Y @ A of Y and A and its completion Y @ A under the n-topology to certain subspaces of % ( A ) . These concepts are discussed in Appendix F, and we shall freely use various definitions and results that are discussed there. In particular, given any g E Y and a E A , the tensor product g @ a is defined in Appendix F3, where it is also pointed out that the mapping (9, a } w g 0a is bilinear. Furthermore, the tensor product Y 0A of Y and A is defined as the span of all such g 0a. Each u E Y 0A has a nonunique representation of the form u = EL=,g k O a k , where g k E Y, a k E A , and r is finite. The function p is defined on any u E Y 0 A by

dU)= inf( k = where the infimum over, the topology The completion of Y @ A is a Banach the form w =

11gkllC ~ ~ ' k ~ ~ , 4= : 1

gk

0a k

I

7

(1)

is taken over all representations of u. p is a norm. Moregenerated by p is called the n-topology (Appendix F5). Y 0A under the n-topology is denoted by Y @ A ; thus, space. Any w E Y @ A has a nonunique representation of g k 0 a k , where gk E 3,a k E A , and m

1

(2)

11gkllC I l a k l l A

k= 1

is finite. Two series and only if

gk

p(

0a k and

gk k= 1

0ak

hk

-

0bk represent the same w E Y @ A if

hk k= 1

8 bk)

(3)

as r and s tend to infinity independently. The value that the norm p assigns to any w E Y @ A is the infimum of the values (2) taken over all representations for w (see Appendix F7). We define a mapping Zof Y @ A into Y ( A ) as follows. Given any w E 9 6A , g k 0a k . Then, define I w by choose any representation w =

2.4.

TENSOR PRODUCTS AND VECTOR-VALUED FUNCTIONS

35

The right-hand side of (4) is a member of 9 ( A ) because the series converges under the norm of % ( A ) by virtue of the finiteness of (2). (Clearly, this convergence is absolute, a fact we shall make use of later on.) We have to show that this definition of Iw is independent of the choice of the representation for w. Let hk 0 b k , where hk E 9 and bk E A , be another representation for w. By virtue of (3), we have, for any e in the dual of 9 0 A ,

1

r

S

k= 1

k= 1

1 e(gk 0 ak) - 1 e(hk 0 bk)

+

0,

r, s

-+

a.

According to Appendix F, Sections F4 and F6, there exists a bijection e w j from the dual of 9 0 A onto the space B(Y, A ) of continuous bilinear forms on 9 x A for which e(gk 0 ak> =j(gk

9

ak)*

Therefore,

for every j E g(9,A ) . In particular, choose j such that j ( g , a) = g(to)q(a) where to is an arbitrarily fixed point of T and q E A‘, A’ being the dual of A . The finiteness of (2) implies that gk(t0)ak and hk(to)bkboth converge in A . Consequently, (5) implies that

1

1

4 ( 1 gktt0)ak) = q ( 1 h k ( r O ) b k )

for all q E A’. But the weak topology of A separates A (Appendix D7), and therefore

1gk(tO)ak = 1hk(tO)bk for every to E T. So, truly, Zw does not depend on the representation used for w. Theorem 2.4-1. The mapping Z: Y @ A - t Y ( A ) dejined by (4) is linear, continuous, and injective. PROOF.The linearity of Zfollows easily from the fact that any representation E 9 6 A and its image gkak in S ( A ) converge absolutely under the norms p and 11 * IlCcA, respectively and therefore can be rearranged (Appendix C 12). To show the continuity of Z,let w E Y 6 A and let gk 0 ak be any one of its representations. S e t f = Zw = gkak. Then,

1gk 0 ak of w

1

1

1

IlfIIG(A)

1 llgkllGllakllA.

36

2.

INTEGRATION WITH VECTOR MEASURES

Since this holds for every representation of w, we have that IlfllccA, I p(w), which implies the continuity of I. We now set about proving the injectivity of I. We have to show that, if (gk}r=l C 9 and {Uk}r=l C A are such that ) ) g k ) )I l a k l I < 00 and f 4 gkak is the zero member of 3 ( A ) , then w C gk 0 ak is the zero member of 9 @ A . That w = 0 E Y 6 A means that, given any E > 0, there exists an equivalent representation w = C hk 0 6, , where hk E 3, 6, E A , and llhkII llbkll < E . We shall prove that such an equivalent representation exists. Let {Ei};= be an arbitrary partition of T, let ti be a point in Ei,and let xi be the characteristic function for E i . Therefore, x i ( t ) = 1 for every t E T. Sincef(ti) = 1 kgk(ti)ak = 0, we may write

1

x

xi

r

1

xi

i=l

m

0

k=l

gk(ti)ak = O.

Now, B O A is equipped with the n-topology, and therefore the mapping k gk(ti)ak converges in A ,

{ g , a } H g 0 a is continuous (Appendix FS). Therefore, since x

Upon interchanging the summations on i and k and subtracting the resulting gk 0 ak E 3 6 A , we obtain representation of O E B @ A from w =

Now,

and (2) is finite. Therefore, given any E > 0, there exists an integer s not depending on the choices of the partition { E i } or the points t i E Ei such that

We now state a lemma but postpone its proof.

Lemma 2.4-1. For any given g E B and { E i } ; = ,of Tsuch that

whatever be the choices of t i E E i .

E

> 0, there exists a partition

2.4.

37

TENSOR PRODUCTS AND VECTOR-VALUED FUNCTIONS

According to this lemma, we can choose a partition {Ei}r= such that, for each k = 1, .. . , s,

Hence, for that partition, (7) and (9) imply that ( 6 ) is the equivalent representation for w that we have been seeking. This completes the proof of Theorem 2.4-1. 0

PROOFOF LEMMA2.4-1. By the definition of 9,there exists a partition {Ei}r= of T and a corresponding simple function i C i X i ,

C i E C , i = l ) ...)r,

i= 1

such that, for every i and every t E E , ,

IAt) - ci I < tea Therefore.

for every t E T. The last two inequalities may be combined to yield (8). 0 We now define some additional notations and terminology that we will be using. The image of 9 6A under the mapping I is denoted by 9 6 A . Thus, 9 6 A is the set of all f E % ( A ) having representations of the form(4)suchthat (2) is finite. Furthermore, the image of 9 0 A under I is denoted by 9 0A . Hence, 9 0 A is the set of all . f ~ ’ 3 ( A having ) representations of the form f = gkak. Thus, for any f E ’3 0 A , f ( T ) is contained in a finite-dimensional subspace of A . For this reason, the members of 9 0A will be called fnitedimensionally-ranging functions. Recall that g0(A) is the space of all A-valued simple functions on T. Clearly,

I;=

9JA) c G OA c 9 6 A .

Corollary 2.4-la. Defne the functional

11. Ill on each f E 9 6 A by

(10)

38

2.

INTEGRATION WITH VECTOR MEASURES

Then, I( * 11 is a norm on Y 6 A. Moreover, Y 6 A equipped with the topology generated by 11.11 is a Banach space, and the mapping I dej7ned by (4) is an isomorphism f r o m Y Q A onto Y 6A . PROOF. By Theorem 2.4-1 and the definition of Y 6 A, I is linear, continuous, and bijective. Therefore, its inverse I - I is also linear and continuous (Appendix D14). Also, for w E Y 6A and,f= Iw and for the norm p defined by ( l ) , we have that 11 f 11 = p ( w ) . Thus, I is an isomorphism, I/ is a norm on Y 0A, and Y 6 A must be complete since Y 6A is complete. 0

Henceforth, it will be understood that Y 6 A possesses the topology generated by (1 . / I l . Y o A is the completion of 9 0A with respect to the norm II . I l l .

2.5. INTEGRATION OF VECTOR-VALUED FUNCTIONS

We are now ready to define the integral of an A-valued function with respect to an [A; B]-valued measure. In this section, P will be an [A; B ] valued measure on K. Consequently, SSVar P < co according to Theorem 2.2-2, and dP, g ( t ) exists as a member of [A; B ] for each g E Y.

Definition 2.5-1. Let f E Y 0 A and choose a n y representation f = E Y and ak E A. We define the integral o f f with respect to p by

cL= gk ak, where gk

To justify this definition, we have to show that the right-hand side of (1) does not depend on the choice of the representation forf. Letf= Cf=lh i b i , where h i E Y and b i E A, be another representation. Now, we can find I linearly independent elements e l , . . , , e , E A such that, for each k and i, I

I

ak =

akjej,

bi =

1P i j e j , k= 1

j= 1

where a k j ,/Iii E C . (See Appendix A3.) Upon substituting these sums into the two representations off and invoking the linear independence of the e j , we obtain Ykakj k= I

=

hipij.

i= 1

2.5

INTEGRATION OF VECTOR-VALU ED FUNCTIONS

39

Hence,

This is what we wished to show. Clearly, then, f - j dP,f ( 1 ) is a linear mapping of Y 0 A into B. Still more is true. It is a continuous mapping because of the inequality

which is established as follows. We know that, for any g

E

9,

Consequently, from (I), we have

5

2 llgkll

Ilakll SSVar p *

Since this holds for every representation off, (2) follows. We can now conclude that the mapping f w J d P ,f(r) possesses a unique extension that is a continuous linear mapping of Y 6A into B. This is how we define the integral dP,f ( t ) on any f E Y 0 A . But, since every suchfis the limit under the norm 11. \I1 of a series g k a , , an equivalent definition is the following. Definition 2.5-2. Let f E Y 6A and choose any representation f

=

cp= gka,, where gk E 9 and ak E A . The integral o f f with respect to P is defined by

It follows that this definition is independent of the choice of the representation for f , that the inequality (2) continues to hold for all f E Y 6 A , and that f ~ d Pj , f ( t ) is a continuous linear mapping of 9 6A into B.

40

2.

INTEGRATION WITH VECTOR MEASURES

In the next two theorems, T and X are two nonvoid sets, 6 and a’ are a-algebras of subsets of T and X, respectively, and p is either a complexvalued measure or a positive measure on C’(Appendix G4). Also, a x 6’ denotes the product a-algebra in T x T (Appendix G3). L l ( p ;A) = L , ( X , a’; p ; A ) is the linear space of Bochner-integrable A-valued functions on Xwith respect to the measure p ; as is explained in Appendix (313, L l ( p ;A) is really a space of equivalence classes of functions, but we speak of i t s members as being individual functions. Finally, 9 ( T x X , a x a’; C ) is the Banach space of all complex-valued bounded functions g on T x X that are the limits of sequences of simple functions under the norm [I * IIc, where Ilgllc 4 sup{ 1g(t,

: t E T , x E XI.

We now state a representation theorem for any f

E

(4)

9 6 A.

Theorem 2.5-1. Let F E L , ( X , a‘; p ; A ) and I E 9 ( T x X , 6 x f(t)

J

X

a’;C ) . Then,

dP,4t, x)F(x)

(5)

exists as a Bochner integral for each t E T and defines a function f where 9 = 9 ( T , (5; C ) . Moreover,

Ilf

111 5

E9

IIFIIL,II 11Ic9

6A, (6)

where

and I p I denotes the total-variation measure of p (Appendix G7). Conversely, every f E 9 6 A has a representation of the form (5).

1“

PROOF. We prove the last statement first. Iff E 9 6A, then f = k = l gkak? where gk E 9, ak E A , and llgkll llakll < 00. Let X = {k}F= let a‘ be the set of all subsets of X , and let p ( E ) be the cardinality of E c a’. [This p is called the counting measure; see Rudin (1966, p. 17).] We can choose an I such that I(t, k) = gk(t) and an F such that F(k) = a k . This immediately yields the representation (5) forf. Conversely, the existence of (5) as a Bochner integral is asserted by Appendix G14. To prove the rest of Theorem 2.5-1, let L , O ( p ; A ) 4 L l o ( X , a’; p ; A ) be the space of all simple functions in L l ( p ;A). Thus, L l o ( p ;A ) is the space of simple functions F = EL=,akXEksuch that, if ak # {0}, then p(Ek) is finite. We define a mapping

1

J : G(T x X , 0. x

,,

a’; C ) x L l 0 ( p ;A ) c-t 9 ( T , a; C ) 0A

2.5 by

'1

rJ(', where again F

=

41

INTEGRATION OF VECTOR-VALUED FUNCTIONS

x)F(x) =

d/Lx

X

k=l

j

dflx / ( f ,

Ek

x)ak

9

I;=akzEk.[That dpx K t , x ) E Y(T, 6 ;C ) 'E,

can be seen by taking a sequence of simple functions on T x X that converges to I and then using the estimate in Appendix G13.1 We next observe that 1141,

F)IIG(A)

'

SUP IET

II[J(k

F,l(t>ll"

1141, a

l l

xex

This shows that the linear mapping F H J ( / , F ) is continuous from Llo(p; A ) , supplied with the topology induced by Ll(p; A ) , into 9 6 A . By virtue of the density of Llo(p; A ) in Ll(p; A ) (Appendix G16), that mapping has a unique extension that is a continuous linear mapping of L , ( p ; A ) into Y 6 A . We can conclude that, for any F € L l ( p ;A ) , the f given by ( 5 ) is a member of Y 6A and that the inequality (6) is a consequence of (7). 0

Our next objective is to develop a Fubini-type theorem. Theorem2.5-2. L e t F E L l ( X , 6 ' ; p ; A ) a n d l E Y ( T xX , 6 x 6 ' ; C ) . Then,

where the outer integral on the left-hand side exists in the sense of Definition 2.5-2 and the outer integral on the right-hand side is a Bochner integral. PROOF. That the left-hand side of (8) exists in the sense of Definition 2.5-2 follows from Theorem 2.5-1. On the other hand, for each x E X , l( ., x ) E Y = Y ( T ,6 ;C ) according to Appendix G14. So, dP, l ( t , x) exists in accordance with Section 2.2. We can choose a sequence {I,,},"=, of simple functions in Y ( T x X , 6 x 6';C ) such that 111 - /,I1 + 0. Therefore, for each x E X ,

2.

42

INTEGRATION WITH VECTOR MEASURES

in [ A ; B]. But, for each n, the left-hand side of (9) is a simple [ A ; B]-valued function of x . Therefore, the right-hand side is a measurable function of x (Appendix G9). Since F E L l ( p ;A ) , we can choose a sequence {F,,}such that F, E L I o ( p ;A ) and F, F in L l ( p ;A ) as well as almost everywhere on X (Appendix G16). Then, for each 11, J T dPl /,,(t, x)F,,(x)is a simple B-valued function of x , and, as n co,it converges almost everywhere on X to -+

-+

Therefore, (10) is a measurable function of x . Moreover, llF(*)llAE L l ( p ;R ) , and

Consequently, by Appendix G15, the right-hand side of (8) truly exists as a Bochner integral. Next, we observe that

according to the inequality (2) of Section 2.2 and Appendix G11. On the other hand, forfdefined by (9,we have t h a t f e S o A. So, by (2) and (6),

5 liFliLlsup I / ( t , .)I

SSVar P.

(1 3)

1, x

The inequalities (12) and (13) and the density of L l o ( p ;A ) in L , ( p ;A ) imply that we need merely establish the equality in (8) for every F e L l 0 ( p ;A ) . So, let F = I akzEI. E L I o ( p ;A ) and let 6‘ E B‘ be arbitrary. We may write

I;=

according to Appendix GI7 and (4) of Section 2.2. But, when # 0, p and b’P( .)akhave finite total variations on Ek and T, respectively (Theorem 2.2-1 and Appendix G7), and therefore we may apply the scalar Fubini theorem to the right-hand side of (14) to interchange the integrations. Then, upon extracting 6‘ and the u k ,we obtain b’ j d P I j d p x I ( t , x)F(x).

2.6.

43

SESQUILINEAR FORMS GENERATED BY PO MEASURES

Since b' E B' is arbitrary, (8) has been established for every F EL I o ( p ;A ) . This completes the proof. 0 Problem 2.5-1. Show that Definition 2.5-1 is consistent with the definition of

f dP,f ( t ) indicated in Problem 2.2-3, for the case where SVar P < 03 and f E 9 0A . [The last two conditions imply that SSVar P < co, according to Problem 2.2-2, and that f

E

%(A).]

2.6. SESQUILINEAR FORMS GENERATED BY PO MEASURES

We end this chapter with a discussion of certain positive sesquilinear forms generated by PO measures. In the first part of this section, P is restricted to being a PO measure on 6. Hence, P maps 6 into [ H ; H I , , where H is a complex Hilbert space with the inner product Moreover, SSVar P = [IP(T)II < co according to Theorem 2.2-3. As usual, E denotes the complex conjugate of any number, function, or measure u. ( a ,

.)a

Lemma 2.6-1. Let f and v be any two members of Y 0 H . Dejne a function 23, on 9 0H x 9 0H by choosing any two representations

where gk , hj E Y and a k , bj E H , and then setting

Then, 8,is a positive sesquilinearform on the space Y 0 H x Y 0 H, and I23P(f,

011

s IIp(m IlflllIlvlll.

(2)

PROOF. An argument like the one following Definition 2.5-1 shows that the right-hand side of (1) is independent of the choices of the representations for f and v. Moreover, 8, is clearly a sesquilinear form on Y 0 H x Y 0 H. To prove the inequality (2), we write

IIP(T)II 2 llgkII k

llhj\l llbjlI.

Ilakll

i

(3)

Since the left-hand side does not depend on the representations for f and v, we may take the infimum over all such representations to get (2).

2.

44

INTEGRATION WITH VECTOR MEASURES

Finally, we show that Bp is a positive form. Let g k , be a simple function that approximates g k , and set r

Then, r

'P(f,f)

-'P(fO,fO)

r

c 1(

=k = l

j=l

IdpIISk(t)G)

-gk,O(t)gj,o(t)lak,

aj).

Through an estimate similar to (3), we see that the right-hand side can be made arbitrarily small by choosing the g k , appropriately. Moreover, functions of the form (4) are themselves simple functions. Thus, to complete the proof, we need merely establish the positivity of 23, on functions of the form fo = a i z E I ,where the Ei comprise a partition of T and are therefore pairwise disjoint. Whence,

and therefore n

SAY0 fo) 9

=

C1 (P(Ei)ai

3

ai) 2 0.

i=

Lemma 2.6-1 has been completely established. 0 In view of Lemma 2.6-1, we can extend 23, continuously onto the Cartesian product 96 H x 96 H supplied with the product topology (Appendix D5).The resulting mapping, which we also denote by B p , will be a positive sesquilinear form on 9 6 H x 9 6 H that satisfies the inequality (2) for all f, u E Y 6 H . We use this result to define still another kind of integral.

Definition 2.6-1. For anyf, u E 4 6 H , we set

I

d(PIf(0, 40)a 2 3 P M v).

(5)

The next theorem presents an explicit formula for ( 5 ) that can be used when representations forfand v are given in accordance with Theorem 2.5-1. The following notation is used. T, X , and Yare three nonvoid sets, &, &', and 6'' are o-algebras of subsets of T, X , and Y, respectively, and 1-1and v are complexvalued measures or positive measures on &' and &", respectively. We know from Theorem 2.5-1 thatf'e 9 6 H if and only if it has the representation

2.6.

SESQUILINEAR FORMS GENERATED BY PO MEASURES

45

where F e L I ( X ,6’; p ; H ) and 1 E Q(T x X,6 x 6’; C ) . Similarly, g E Q 6 H if and only if v(t) =

f dv, m ( t ,Y ) V ( Y ) , Y

(7)

where V € L 1 (Y , 6”;v ; H ) and m E S(T x Y , 6 x 6 ” ; C ) .

Theorem 2.6-1. Let f and u have the representations ( 6 ) and ( 7 ) . Then,

PROOF.We first note that the mapping ( 4 x , Y ) H l ( t ,x ) m ( t ,A is a member of Y(Tx X x Y , 6 x 6’ x 6 ” ; C ) . This fact and a straightforward estimate using ( 2 ) of Section 2.2 show that the right-hand side of (8) exists as a scalar integral. For the integrable simple functions r

=

1 akX& E L pcx,

k= 1

6 ‘ ;p ; H ,

and S

V

=

1 bjXI, E Llo(Y, 6 ” ; V ; H ) , j= 1

the right-hand side of (8) is equal to

Since p , V, and ( P ( ‘ ) a k ,b j ) have finite total variations on E k r I j , and T, respectively, whenever ak # 0 and b j # 0, we may apply the scalar Fubini theorem and then extract the ak and bj to obtain

Thus, (8) is true for any integrable simple functions F a n d V.

46

2.

INTEGRATION WITH VECTOR MEASURES

To show that (8) remains true for all F E & ( X , 0'; p ; H ) and

v E L,( Y , (5"; v ; H ) , we need merely show that both of its sides depend continuously on F and V with respect to the L , norms (Appendix D5). For the left-hand side, this follows from

5 IIP(T)II IIFIIL, SUP I l ( t ,

x>l II

f , X

w,

SUP f,

I m ( t , Y)l.

(9)

Y

It is easily seen that the right-hand side of (8) is also bounded by the righthand side of (9). 0

In the remainder of this chapter, P : Km -+ [ H ; H I , is a a-finite PO measure. Also, we let g E Y, g(t) > 0 for all t E T, and g be integrable with respect to P (Definition 2.3-2). Moreover, we set

Q ( E )4

I

E

dPfg(t>,

By Theorem 2.3-3, Q is a PO measure on Yg(H)4 Y,(T, (5; H )

(5.

E E 0.

Finally, we set

{ f e Y(H):f/g E 9 6H } .

(10)

Lemma 2.6-2. Y g ( H )c 9 d , ( H ) c Y 6 H .

PROOF.We will use the obvious fact that, if u E 9 and q E Y 6 H , then uq E 9 6 H . Since g E 9, Jg E 9 also. By definition, for any f E Y,(H), we have thatflg E 9 6 H . Therefore, Jgflg =f / J g E Y 6 H , which implies that Yg(H)c Yd;(H).Another multiplication by Jg shows that F?dq(H) c Q 6 H. 0 Lemma 2.6-3. Iff E 9 , ( H ) , thenf has the representation ( 6 ) where, in addition, thefunction { t , x } ~ l ( tx ), / g ( i ) is a member o f Y ( T x X , (5 x (5'; C). This lemma follows directly from Theorem 2.5-1. Definition 2.6-2. For any f E Y g ( H ) we , set

Note that the right-hand side has a sense according to (10) and Definition 2.5-2. Clearly, f~ d P , f ( t ) is a linear mapping of Yg(H)into H. Moreover,

2.6.

47

SESQUILINEAR FORMS GENERATED BY PO MEASURES

this definition of dPtf ( t ) is independent of the choice of g. To see this, first note that, for every f E Q,(H) n Q 0H , we may choose a representation, f= gka,, where a, E H, g k E 3,and lgk(t)I < Ckg(t) for all t E T, the ck being constants. [Indeed, sinceflg E Y 0H , we can writeflg = qkak and choose ck supt Iqk(t)l.] Therefore,

I;=

According to Theorem 2.3-2, the right-hand side does not depend on the choice of g. Next, it can be seen as before that every f E Q g ( H ) has a representation of the form f = g k ak, where ak E H , g k E 9,1gk.t) I < Ckg(t) for all t E T, the ck are constants, and Cp=lCkllakII < 00. Set f , = gkak. As r + 00,

I;=l

in H because

Since the left-hand side of (1 1) does not depend on the choice of g, neither does the right-hand side.

Definition 2.6-3. For anyf, v E Q,(H), we set

In view of Definition 2.6-1 and Lemma 2.6-2, the right-hand side has a sense and defines a sesquilinear mapping on Y,(H) x %,(If). Moreover, (12) is independent of the choice of g, as can be seen through an argument similar to the one used for Definition 2.6-2.

Theorem 2.6-2. Let f E 9 , ( H ) and v E Yg(H) have the representations ( 6 ) and (7). Then, (8) of Section 2.5 and (8) of this section still hold true in the present situation, where P is a a-jinite P O measure.

PROOF.By Definition 2.6-2,

48

2.

INTEGRATION WITH VECTOR MEASURES

By virtue of Lemma 2.6-3, Theorem 2.5-2, and Theorem 2.3-4, the righthand side is equal to

and this justifies (8) of Section 2.5. A similar manipulation establishes the other equation. 0

Problem 2.6-1. Show that the definition of b, given by (1) does not depend on the choices of the representations forfand u. Problem 2.6-2. Show that (12) does not depend on the choice of g. Problem 2.6-3. Prove the other part of Theorem 2.6-2. Problem 2.6-4. Let P be a a-finite measure and let g E 9 be a positive function [i.e., g(t) > 0 for all t ] that is integrable with respect to P. Furthermore, IetS, u E gg(H)and let {Ek}:=l be an increasing sequence in 0 with UEk = T. Show that

and

Chapter 3

Banach-Space-ValuedTesting, Functions and Distributions

3.1. INTRODUCTION

As was mentioned in the Preface, the natural framework for a realizability theory of continuous linear systems is distribution theory. Since the signals in the systems of concern to us take their values in Banach spaces, the properties of Banach-space-valued distributions are essential to our purposes. The present chapter is devoted to a discussion of such distributions; they constitute a special case of the vector-valued distributions of Schwartz (1957). We start with a description of the primary testing-function space in the theory of distributions, namely 9 ” ( A ) . As always, A and B denote complex Banach spaces.

49

3.

50

BANACH-SPACE-VALUED DISTRIBUTIONS

3.2. THE BASIC TESTING-FUNCTION SPACE 9 " ' ( A )

Let nz be an n-tuple each of whose components is either a nonnegative ) the integer in R' or co. Also, let K be a compact set in R".9 K m ( A denotes linear space of all functions 4 from R" into A such that supp 4 c K and, for every integer k E R" with 0 2 k m, 4(k)is continuous. We assign to g K m ( A )the topology generated by the collection { y k : 0 I k I m} of seminorms, where

Since y o is a norm, g K m ( A is ) separated. Moreover, it is metrizable because the collection { y k } is countable (Appendix C7). Lemma 3.2-1.

9Km(A)

is complete and therefore a FrPcliet space.

PROOF.Since g K m ( A )is metrizable, we need only establish its sequential completeness (see Appendix CI1). Let {4i}?=, be a Cauchy sequence in g K m ( A )In . view of (1) and the completeness of A , we have that, for every k as restricted above, there exists an A-valued function $k on R" for which 4!k)--t $ k uniformly on R". By note VlII of Section 1.4, $ k is continuous. Also, by Problem 1.6-3, $bk) = $k . Clearly, supp t,b0 c K. Hence, $o is the ) {+i}. O limit in g K m ( A of

Note that, if every component of m is finite, g K m ( A )is a Banach space because its topology is the same as that generated by the single norm p , where

P(4)

max

Yk(4).

O$k$m

When all the components of m are

00

(i.e., when m

=

[a]), we denote

g K m ( Aby ) a K ( A ) .Moreover, we set g K m ( C= ) gKm and 9 K ( C )= g K .

Now, let { K j } Y = , be a sequence of compact sets in R" such that Kl c K2 .* K j = R",and every compact set J c R" is contained in some K j . We define g m ( A )= 9:,,(A) as the inductive-limit space generated by the 9 E j ( A ) . That is, c K3 c

9,

uj

u9 g j ( A ) , m

9"(A) = 9 l " ( A )=

k= 1

and this space possesses the inductive-limit topology (see Appendix El). As before, we set 9[m1(A) = B(A),Bm(C)= 9", and 9 ( C ) = 9. This definition of g m ( A )does not depend on the choice of { K j } .Indeed, for any other sequence {Hi}zl of compact sets with the required properties, 9 E j ( A ) and 9 G i ( A ) are identical as linear spaces because every 9 F j

u

u

3.2.

THE BASIC TESTING-FUNCTION SPACE

9"'(A)

51

is contained in some 9zi and conversely. To show that the topologies are 9z,(A). Given any K j , the same, let A be a convex neighborhood of 0 in we can find an H i containing K j . By definition of the inductive-limit topology, A n 9 E i ( A )is a neighborhood of 0 in 9 z , ( A ) .Moreover, 9 F j ( A )is a subspace of 9 i i ( A ) , and its topology is the same as that induced on it by 9;,(A) because both topologies are generated by the same seminorms y k . Hence, An is a neighborhood of 0 in 9E,(A). Consequently, A is a convex 9 F j ( A ) . Similarly, every convex neighborhood of neighborhood of 0 in 0 in 9 E j ( A ) is a convex neighborhood of 0 in 9 i , ( A ) . Consequently, the two inductive limit topologies are identical 9"'(A) is clearly a strict inductive-limit space (Appendix E3). Moreover, it possesses the closure property defined in Appendix E4 because each 9g4(A) is complete. As a consequence, the following assertions hold. 9"'(A) IS a complete separated locally convex space. A linear mapping f of 9"'(A) into another locally convex space W is continuous if and only if its restriction to each 9 E j ( A ) is either sequentially continuous or bounded. A set is bounded in 9"'(A) if and only if it is contained and bounded in some 9 $ ( A ) . Similarly, a sequence { 4 j }converges in W ( A ) if and only if it is contained and converges in some 9 g j ( A ) .Thus, a linear mapping on W ( A )is continuous if and only if it is sequentially continuous. (See Appendix E.)

ui

uj

u

ui

Lemma 3.2-2. Let J and Kbe two compact intervals in R" such that J contains a neighborhood of K . Then, given any 4 E gK"'(A),there exists a sequence {4j}y= c g J ( A )such that $ j + 4 in 9,"'(A).

PROOF.Set

and 9

P

= 1,

2, . . * .

(In this proof, all integrations are over R".) It follows that q, is a smooth nonnegative function, diam supp q p = 2/p, and j q,(t) dt = 1. Next, set

For all sufficiently large p , supp 4, c J . Moreover, we may differentiate under the integral sign (Theorem 1.6-4) and integrate by parts (Problem 1.6-1) to obtain, for any fixed k E R" such that 0 Ik I m,

4r)(t)= J$(k)(x)vp(t - x) d x .

3.

52

BANACH-SPACE-VALUED DISTRIBUTIONS

Hence,

sup

l14(k'(t)- 4 ' k ' ( X ) I I A *

lr-xl
By the uniform continuity of 4(k)on R" (Theorem 1.3-1), the right-hand side tends to zero uniformly for all t as p m. Thus, {4p}with a sufficient number of initial terms deleted is the sequence we seek. 0

Lemma 3.2-3. 9 ( A ) is a dense subspace ofQ"(A). Let 4 E 9"'(A) be given and choose arbitrarily a neighborhood A in 9 " ( A ) . Also, choose the compact intervals J and K such that supp 4 c R c K c 3. Then, A n g J " ( A ) is a neighborhood of 4 in 9 j " ( A ) and, that ) is contained in by the preceding lemma, we can find a I) E ~ ~ ( A A n 9,"(A). Hence, every neighborhood A of 4 in W ( A ) contains a I) E 9 ( A ) . The last statement is equivalent to our lemma. 0 PROOF.

of

4

3.3. DISTRIBUTIONS

An [ A ; B]-valued distribution f on R" is by definition any continuous linear mapping of 9 ( A ) into B ; i.e., f E [ 9 ( A ) ;B ] . It will be shown in Section 3.5 that f can be identified with a unique continuous linear mapping of 9 into [ A ; B]. This is the reason for callingf" [ A ; B]-valued." The canonical injection of 9 ( A ) into 9 " ( A ) is clearly continuous for every m. Consequently, the restriction of anyfE [9"'(A);B] to 9 ( A ) is a member of [ 9 ( A ) ;B], and this restriction uniquely determinesf because of the density of 9 ( A ) in 9 " ( A ) . Thus,

[9"(4; BI c [%I); BI. In the same way, we have that [Q"(A);BI = [BP(A); BI whenever m I p . Upon setting B = C, we obtain the dual [ 9 " ( A ) ; C] of 9 " ( A ) . Similarly, with A = C , we get a space [9";B] of B-valued distributions. Note that [C; B] can be identified with B so that [9"; [C; B ] ] and [9"; B] can be considered to be the same space. Finally, when A = B = C and m = [ a ] , [ 9 " ( A ) ; B] becomes the customary space [9;C ] = 9' of all complex-valued distributions.

3.3.

53

DISTRIBUTIONS

Lemma 3.3-1. Given an f E [9"'(A); B] and a compact interval K c R", there exists an integer p E R" with 0 5 p m and a constant M > 0 such that for all 4 E 9K"'(A),

11 (f, 4) 11 B 5 M p p ( 4 ) ,

(1)

where

Pp(4) 4L Omax
SUP

I14'k'(t>llA.

IEK

M andp depend in general on f and K.

This lemma follows directly from Appendix D2 and the fact that the linear mappingfis continuous on g r n ( A )if and only if its restriction to every 9 K m ( A ) is continuous. Unless something else is explicitly indicated, we always assign to [9"'(A);B] the bounded topology. This is the topology generated by the collection {y@}@ of seminorms, where 6 denotes the set of all bounded sets in W ( A ) and

ydf)

'

11 ( f 4 ) 11 B .

(3)

9

+E@

We also refer to the bounded topology as the 6-topology. On occasion, a weaker topology is assigned to [9"'(A);B ] , namely the pointwise topology. It is generated by the collection { y + } of seminorms, where 4 traverses 9"'(A) and Y+(f>

II(f,

(4)

4)IIB.

To indicate that the pointwise topology is being used, we employ the notation [ 9 " ' ( A ) ;B]". Both the bounded and pointwise topologies are separating.

Lemma 3.3-2. Every f through the equation

E

[ P ( A ) ;B] uniquely defines a g

(9, 0)a

(f, Oa),

E

[9"'; [ A ;B ] ]

0 E grn,a E A.

(5)

This assertion remains true when 9"'(A) is replaced by 9 K " ' ( A ) and 9"'by gK"' for any given compact set K.

PROOF.Having fixed upon some 0 E B"', we define a mapping,je of A into B by j,a A (f, Oa) for all a E A . It readily follows that j e is linear. That it is also continuous follows from Lemma 3.3-1. Indeed, for any compact set K that contains supp 6' and for all a E A , we have Iljeallr, =

II(fe9

0a)IIB 5 MpAOa) 5 M l l a l l A ~ ~ ' ( O h

54

3.

BANACH-SPACE-VALUED DISTRIBUTIONS

where

~ ~ ' (=0 )max sup 1 O c k ) ' ( t ) l . O s k s p tsR"

Hence, IljOll[A: B ]

(6)

Mpp'(0).

Next, set ( g , 0 ) j,. This uniquely defines g as a mapping from 9"into [ A ; B ] . g is linear because, for any a E A , a, p E C, and 0, $ E grn,

+ P$>a = (.A

( 9 , a0

+ P$a>

= a(ft =

ea> + K.A $4 + P ( g , $>)a.

( 4 g ,0)

Moreover, (6) implies that g is continuous. The second statement of the lemma is established in the same way. 0

For any nonnegative integer k

E

R", we define the generalizeddrflerentiation

Dk on [ 9 " ( A ) ;B] as follows. First, note that Dk is a continuous linear mapping of 9 m + kinto ( ~g), " ( ~ since ) its restriction to every 9 z + k ( A )is a continuous linear mapping of 9 : + k ( A ) into a K r n ( A )This . allows us to define Dk on any f E [ g r n ( A )Bl ; by

( 0 %4)

A (-

l ) l k l ( . A Dkb),

6 E 9rn+k(A)

(7)

because the right-hand side has a sense. Theorem 3.3-1. Dk is a continuous linear mapping of [ g r n ( A ) B] ; into [ 9 r n ' k ( A ) B], ; as wellasof [ 9 " ( A ) ;B]" into [ 9 r n ' k ( A ) B]". ; PROOF.Observe that the right-hand side of (7) is a member of B, so that ) B. It follows readily that D Y E [ 9 m + k ( A )B; ] and that Dkfmaps 9 r n + k ( Ainto Dk is a linear mapping of [ g r n ( A ) B] ; into [ 9 m + k ( A )B; ] . To show that Dk is continuous, let @ be a bounded set in 9 " ' + k ( A ) .As 4 traverses @, Dk4 traverses a bounded set in 9 " ( A ) ; call it 0.So,

supll(D% + € @

4>llB

= suPll(f,

Dkb)llB

4EO

= supll(f,

O>llB*

So, truly, Dk is continuous from [ 9 " ( A ) ;B] into [ 9 m ' k ( A ) ;B ] . By restricting @ to sets of one element each, we obtain the same conclusion with the pointwise topologies. 0 Another operator of importance to us is the shifting operator ur, which is also called the translation operator. Let T E Rn be fixed. 6, is defined on any

3.3.

DISTRIBUTIONS

55

4 E 9 " ' ( A ) by (a,4)(t)= 4(t - z). It is an automorphism on 9"'(A). On the other hand, a, is defined on any f E [9"(A);B] by (arf,

4)

'( f ,

0-r

4),

4 E g"(A)*

(8)

As a consequence, a, is an automorphism on [ 9 " ( A ) ; B], as well as on

[.9"(A); B]". (Show this.) Two distributions f, g E [ 9 ( A ) ; B] are said to be equal on an open set SZ c R" if (f, 4 ) = ( g , 4 ) for every 4 E 9 ( A ) such that supp 4 c a. The null set o f f is the union of all open sets on which f is equal to zero (i.e., is equal to the zero distribution). The complement of the null set is called the support off and is denoted by suppJ Thus, suppf is a closed set. A property of distributions that we occasionally use is the following. If f E [ 9 ( A ) ;B] is equal to zero on every set in a collection of open sets, then it is equal to zero on the union of these sets. The proof of this assertion is precisely the same as it is for complex-valued distributions on 9 and can be found, e.g., in the work of Zemanian (1965, Section 1.8). One consequence of this result is that supp f is the smallest closed set outside of which f is equal to zero. Another direct result is the following. (f, 4 ) depends only on the values that 4 E 9 ( A ) assumes on any arbitrarily small neighborhood of suppJ Indeed, if $ E 9 ( A ) is equal to 4 on a neighborhood of supp f, then supp(4 - @) is contained in the null set off, so that (f, 4 - $) = 0. We conclude this section with two examples of [ A ;B]-valued distributions. Example 3.3-1. Let F be a fixed member of [ A ; B] and let S be the delta function defined on any 4 E .9'(A) by (6, 4 ) 4(0). We define FS as a mapping on G'(A) by (Fa, 4 )

e F4(0),

4 E 9'(4.

(9)

Consequently, FS E [ 9 ' ( A ) ; B]. As is indicated in Lemma 3.3-2, F6 generates a member of [ g o ;[ A ; B]], which we will also denote by FS. Thus, in accordance with ( 5 ) and (9), we write, for any d E 9' and a E A , (FS, O)a = (Fa, Oa)

= F[O(O)a]= [Fd(O)]a

or (FS, 0)

= Fd(0).

Upon applying the generalized differentiation Dk to FS, we obtain, for any

4 E .9k(A), ( Dk(FS),4 ) = (- l)'k'F4'k'(0) = F( Dk6, 4).

(10)

We define FDkS as the distribution that assigns to 4 the value indicated on the right-hand side of (10). Thus, we have Dk(FS) = FDkS. 0

56

3.

BANACH-SPACE-VALUED DISTRIBUTIONS

Example 3.3-2. Let h E L , ( [ A ;B]); that is, the [ A ; B]-valued function h is Bochner integrable with respect to Lebesgue measure on the Borel subsets of R". We define a mapping f of 9'(A) into B by setting

By Appendix G11,

II(L 4)llB

I141Ll SUP II4(t)ll,. 1

It follows that h H f is a continuous linear mapping of & ( [ A ; B]) into [go@); B]. This mapping is injective; indeed, if / I generates the zero member of [ 9 ' ( A ) ; B], then JRn

h(t)O(t)dt a

=0

for all 0 E 9 and all a E A , which by Appendix G12 implies that h ( t ) = 0 for almost all t . Thus, L , ( [ A ; B]) can be identified as a subspace of [9 '( A ) ; B]. Any member of [ 9 ' ( A ) ; B] that can be generated in this way from a member of L,( [ A ; B]) will be called a regular [ A;B]-valued distribution. 0

Problem 3.3-1. Show that the shifting operator oris an automorphism on [ 9 Y A ) ;BI. Problem3.3-2. Let h be an [H; HI-valued function on R with the following two properties. It is strongly measurable with respect to Lebesgue measure on the Borel sets; that is, h(*)ais a measurable H-valued function for every a E H. It is locally essentially bounded; that is, for every compact set K c R , there exists a constant MK such that Ilh(t)II 5 MKfor almost all t E K. Define a mapping f on any 4 E 9' by

(f, 4)a

a / h ( t ) U 4 ( l ) dt.

Show that f E [9';[H; HI].

Problem 3.3-3. Let P be a a-finite [ A ; B]-valued measure on the bounded Borel subsets of R". For every 4 E 9 ( A ) ,define f by

(L4) a p p 4 . Show that f E [ D ( A ) ;B]. f is called the [ A ;B]-valued distribution generated by P.

3.4.

57

LOCAL STRUCTURE

3.4. LOCAL STRUCTURE

The objective of this section is to show that every Banach-space-valued distribution can be represented on any compact interval as a finite-order derivative of a continuous Banach-space-valued function. We do this by means of a method employed by Sebastiao e Silva (1960). As before, K is a compact interval in R", and m = {mi}:= denotes an n-tuple each component of which is either a nonnegative integer or co. %?,"(A) is the linear space of all functions 4 from K into A such that, for each integer k E R" with 0 I k 2 M , dCk)is continuous. %?,"(A) is equipped with the topology generated by the collection { Y k : 0 5 k 2 m > of seminorms, where again Yk(4)

suPl14'k'(t)llA. 1 E K

%?Kn'(A) is a FrCchet space, its completeness being established as was Lemma 3.2-1. When every component of m is finite, %?,"(A) is a Banach space. By identifying each t,b E 9,"(A) with its restriction to K , we' can and will view 9,"(A) as a subspace of %?,"(A). Now, let p = {pi):=l be a nonnegative integer in R", and let [L, Q] be a compact interval in R", where L = { L i } : = ]< Q = {Q,}Y=,. It is a fact in interpolation theory that, for each i, there exist 2(pi + 1) polynomials on [ L i , Qi], which we denote by g x i , v i , where X i = L i , Qi and vi = 0, . . . , p i , such that df'g,,, v,(Li)

= dvi.pi

a f i S l , ,v i ( Q i ) = 0,

9

afigQi, v,(Li)

= 0,

d;'gQi, vi(Qi)

= dvi,pi

*

Here, p i = 0 , . . . , p i and d v i , p i is the Kroriecker delta (i.e., dvi,,li = 1 for v i = p i and d v i , p i = 0 for vi # pi). Now, consider the function d b obtained from some 4 E gKP(A) by means of the following formula:

A summation on the X's means the sum of all possible terms obtained by is a summation over setting each X , equal to either L, or Q,. Also, 1s i s n ,

0 1 vi_
3.

58

BANACH-SPACE-VALUED DISTRIBUTIONS

C2 is a summation over 1
Finally,

i+ 1I j s n ,

0 s vi
01 v j < p j .

Enis a summation over 0I \'I I p l ,

0 < \ ' 2 I p * , ... ,

0I V" I P " .

Lemma 3.4-1. (i) For euery integer k E R" hlith 0 k 5 p , 4Lk)(t)= q!~'~'(t) f o r all t on the boundary of K . (ii) 4 H 4 b is a continuous linear mapping o f g K P ( Ainto ) gKP(A). PROOF.That 4,, E V K P ( Afollows ) from the facts that each g x i ,y i is a polynomial i n t i alone and 4 E %',,(A). 'The assertion (ii) follows directly from the definition of 4 b . Finally the assertion (i) can be verified through some straightforward (but tedious) computations. 0 A n operator F that maps a space V into Y is called a projection if FF4 F 4 for all 4 E Y .

=

Lemma 3.4-2. The operator np: (b H4 - 4 b is a continuous linear projection of V K p ( Aonto ) QKP(A). PROOF.According to Lemma 3.4-1, 4 - ( b b E BKp(A). (It is understood here that 4 - 4,, is extended onto the exterior of K as the zero function.) That lemma also implies that nP is a continuous linear mapping of %',,(A) into %,"(A). But Q K P ( A )is a subspace of g K p ( A and ) possesses a topology that is identical to the topology induced on it by % / ( A ) . Therefore, np is a continuous linear mapping of K K P ( A into ) g K p ( A ) Finally, . the definition of 4 b also shows that, if 4 E Q K P ( A ) ,then 4b is the zero function in WKP(A).Consequently, np4 = 4 for all 4 E B K p ( A )so , that npnpd)= n P 4 for all d, E CeKP(A). That is, n p is a projection. 0 Next step: We set up the function

Here, 1 + denotes the unit-step function on the real line; that is, 0,

ti < 0

1,

t i > 0.

For each fixed t E R", we denote the function X H J,(f - s)on K by J,(t - .), Also, we remind the reader that the symbol [2] denotes the it-tuple each of whose components is equal to 2.

3.4.

59

LOCAL STRUCTURE

Lemma 3.4-3. Let 4 E g f ; + [ 2 1 ( A )Then, . t H J,(t

- .)DP+[Z]4 ( t >

is a continuous mapping of R" into %Kp(A).Similarly, r HJ,(t - .) is a continuous mapping of R" into VKp. PROOF.Let the integer k E R" be such that 0 I k d p . Also, fix t and let I be any compact interval in R" containing t in its interior. As a function of { t , 4,

D,kJ,( t - X)D,P + [2lf$( t ) is a continuous function from I x K into A , and therefore it is uniformly continuous on I x K. So, as At -+0 in R",

D,kJ,(t

+ At - X ) D P ' " ~ ~+( ~At) + D,kJ,(t

- x)DP"']q5(t)

in A uniformly with respect to all x E K. This proves our assertion for the function (1). The proof for the other function is the same. 0

Lemma 3.4-4. For any

4 E 9i+[z1(A),

Note. I n the right-hand side, we have the Riemann integral of the continuous %'KP(A)-valued function (1).

PROOF.We may repeatedly integrate by parts to establish that (- l)IpI

s

J,(t - x ) D p + " l ~ ( tdt ) =

K

s

Jo(t - x)DcZ1q5(t) dt

= q5(x).

K

Equation (2) follows directly. 0 We now appeal to note 11 of Section 1.4 and to Lemma 3.4-2 in order to apply 7 c p to (2) under the integral sign. Thus, for any E 9i+[21(A),

where G,(t, *) = n,J,(t - -).

The integral in (3) is a continuous gKP(A)-valued function of t E K because it is the result of first applying the function (1) and then applying x p , both of which are continuous mappings. Similarly, G,(t, -) is a continuous gKPvalued function o f t E K.

60

3.

BANACH-SPACE-VALUED DISTRIBUTIONS

Lemma 3.4-5. Let f E [9Y'(A); B], let K and J be compact intervals in R" such that K c 3, and let p be the integer in R" corresponding to f and J in accordance with Lemma 3.3-1. Then, there exists a unique 1 E [ 9 K p ( A ) ;B] that satisjies the relation (1, O )

=

lim ( f , O j ) j-+m

(4)

for all O E .9KP(A) and all sequences {e,} c .9j"'(A) that tend to 0 in 9 j p ( A ) . Thus, (1, $) = ( f , $)for all $ E 9 K m ( A ) .

PROOF.Choose any 8 E 9 K p ( A ) .By Lemma 3.2-2, we can choose a sequence {O,}j", c gJ'"(A)such that O j + 0 in 9 j p ( A ) . We infer from the inequality in Lemma 3.3-1 that {( f , 0 , ) ) is a Cauchy sequence in B. Its limit is taken to be the value (1, O), and this value also satisfies the inequality in Lemma 3.3-1. It follows that 1 E [.9KP(A);B]. Moreover, this definition of 1 is independent of the choice of the sequence {O,}, and 1 is uniquely determined by the values thatfassigns to the 4 E 9j"'(A). 0 Theorem 3.4-1. Let f E [ W ( A ) ; B] and let K be a compact interval in R". Then, there exists an integer p E R" with 0 p _< m and a continuous [ A ;B]valuedfunction h on Ksuch that,for'all4 E .9:'[21(A),

I

(f,4 ) = h(t)DP+['l4(t) dt. K

(5)

In general, p and It depend on,f and K. PROOF.Let J and p be chosen as in Lemma 3.4-5. Then, 1 can be applied to (3), where now 4 E 9F'[21(A), and note 11 of Section 1.4 can be invoked to get

Let g E [gKP; [ A; B]] be related to 1 in accordance with Lemma 3.3-2. Then, (6) can be rewritten as

Upon setting h(t) = (- l)Ip1(g(*), G,(t, we arrive at (5). h is a continuous [ A ; B]-valued function because it is the composite of the two continuous mappings G,(t, .): K - +Q K P and 9: 9 K p -+ [ A ; B]. a))

That p and h depend on f and K can be shown by examples. 0

3.5. [ B ( A ) ;B] AND [9; [A; B ] ] CORRESPONDENCE

61

Corollary 3.4-la. Let f E [gm; A ] and let K be a compact interval in R". Then, there exists an integer p E R" with 0 p I m and a continuous A-oa1uedfiitictio.n h on Ksuch that, for all 4 E BE'[21,

(f,4 ) =

1

h ( t ) D P + [ 2 1 4 ( dt )t .

(7)

K

Here, too, p and h depend in general onf and K.

PROOF.By Theorem 3.4-1, (7) holds true for a continuous [ C ; A]-valued function h on K. But [ C ;A ] is identical to A. 0 Problem 3.4-1. Supply the details of the proof of Lemma 3.4-1. Problem 3.4-2. Show that the quantities p and I? in Theorem 3.4-1 depend in general on f and K.

3.5. THE CORRESPONDENCE BETWEEN [ 9 ( A ) ; B], AND 19; [ A ; Bl1

The natural identification between [ 9 ( A ) ;B ] and [9; [ A ; B ] ] ,which we alluded to at the beginning of Section 3.3, will now be established. In the following, B O A (or B K O A ) denotes the linear space of all 4 ~ 9 ( A ) [respectively 4 E g K ( A ) ]having representations of the form 4 = 0, a,, where 8, E 9 (respectively 0, E 9,), ak E A , and the summation is over a finite number of terms.

1

Lemma 3.5-1. Let K and N be compact interoals in R" such that K c fi. Given any 4 E 9 K ( A ) ,there exists a sequence {4j}T= c BN0A that converges in B N ( A )to 4. Thus, 9 0A is dense in 9 ( A ) . PROOF.

Let L be a compact interval in R" such that K c L and L c fi. Let

d, = inf{ I t - X I : t E K , x E R"\L}, d2 = inf{ 1 t - XI : t E L , x E R"\N). Thus, d , and d2 are greater than zero. We choose a sequence {QP},"=,, where each Ep is a collection { Q p , i } i of open sets in R" with the following three properties. Only a finite number of the Q p , intersect any bounded set in R". For every p and i, diam QP, < d l . Finally, sup diam QP, 1

+ 0,

p

00.

3.

62

BANACH-SPACE-VALUED DISTRIBUTIONS

Next step: For each p , we can choose a collection {I),, i } i c 9 such that, for every i and 1, we have supp I),, c Q,, i , 0 I I),,,i ( t ) I 1, and I),, i ( t ) = 1. [See, for example, Zemanian (1965, Section 1.8).] For each p , we define the mapping J , on 9K(A)into gL0A by

xi

where each t i is chosen as some member of 0,. i . It follows that, as p J p 4 tends to # in g L O ( A ) .Indeed, su~II+(t)- C 4 ( t i ) I ) p , i ( t ) I I A I SUP i

t

t

4(ri)III)p,i(f)

i

5 SUP i

1 II+(t> -

+ co,

SUP

II4(t) - +(ti)Il>

t€Rp,i

and the right-hand side tends to zero as p + co because of the uniform continuity of # E 9K(A). Next, let q,, where q = 1 , 2 , . . . , be the function defined in the proof of Lemma 3.2-2, and let R , be the linear mapping on gO (A)defined by

9-

1

9(x)q,(* - X) dx.

R"

If q-' < d,, then supp R,9 c N whenever supp 0 c L. Moreover, R , is continuous from g L O ( A ) into BN(A). Indeed, for any O€BL0(A) and any nonnegative integer k E R",

supllDtkR,O(r)ll =

e(x)D,kq,(r - X) d x

t

Now, for any 4 E g K ( A ) , consider R , J p 4 E gN 0A. By what we have already shown, for fixed q > d; ' and as p --+ co, we have that R , J,, # --t R , 4 in BN(A).Furthermore, by the proof of Lemma 3.2-2, R 4 4 + 4 in 9N(A) as q + co. The fact that 9 N ( A ) has a countable base of neighborhoods of 0 (Appendix C7) can now be exploited to extract a sequence { R ( , , J p j $ } 7 = , that converges in 2 , , , ( A ) to 4. This establishes the first conclusion. The second conclusion follows from the definition of 9 ( A ) as an inductivelimit space.

Lemma 3.5-2. 9 0A is dense iii 9'"(A) wliatewr be m.

3.5. [ g ( A ) ;B] AND [9;[ A ; B]]CORRESPONDENCE

63

PROOF.9 ( A ) is a dense subspace of 9"'(A), and the topology of 9 ( A ) is stronger than that induced on it by 9 " ( A ) . Consequently, this lemma is an immediate result of the preceding one. 0

Theorem 3.5-1. There is a bijection from [ 9 ( A ) ;B] onto [9;[ A ; B ] ] defined bY

(9,O a = ( f , @a>,

(1)

where 0 E 9, a E A , g E [9; [ A ; B ] ] ,and f E [ g ( A ) ;B]. Note. Because of this result, we will subsequently denote g and .f by the same symbol, sayf, and will replace (1) by

(2)

( f , @>a= ( f , ea>.

PROOF.Lemma 3.3-2 shows that f uniquely determines g though (1). So, let us consider the converse. By Corollary 3.4-la, for any given g E [9;[ A ; B ] ] and compact interval K c R", there exists a nonnegative integer q E R" and a continuous [ A ; B]valued function / I on K such that, for all O E g K , (9,8 ) =

j

h(t)Dq8(t)dt.

K

(3)

We let 9 k ( A ) be the space of all 4 E 9 ( A ) whose supports are contained in and supply with the topology induced by g K ( A ) .Also, 9~ 9 g ( C ) . We define a mapping,fX on g i e ( A )into B by

ICI)

(fie,

j h(W41CI(t)d t , K

ICI E %(A).

(4)

It follows that,f, E [ 9 , ( A ) ; B ] . Moreover, if we set = 8a with 8 E and we see from (3) and (4) that (9,%)a = (f k , Oa), in agreement with (1). This procedure determines an f g E [Qk(A); B] for every compact interval K. We now assert that, if the compact interval J contains K,then the restriction of,fi to 9 k ( A ) coincides withf, . Indeed, for 8 E g kand a E A , (fJ, Oa) = (g,8)a = ( f g , %a), and therefore coincides with fk on 9 k 0A . But, as a consequence of Lemma 3.5-1, gg0A is dense in 9&), and this implies our assertion. It now follows that there exists a unique f~ [ 9 ( A ) ;B] whose restriction to each 9 k ( A ) coincides with,fk. Thus, every g E [9;[ A ; B ] ] uniquely determines anfE [ 9 ( A ) ;B] by means of (1). 0 a E A,

fJ

64

3.

BANACH-SPACE-VALUED DISTRIBUTIONS

3.6. THE p-TYPE TESTING FUNCTION SPACES

The rest of this chapter discusses various generalized-function spaces, most (but not all) of which are subspaces of [B"(A);B ] . It is possible to formulate our discussion in such a general fashion that each space of interest to us becomes a special case obtained by making particular choices for certain parameters. The only exceptions to this are the spaces of L,-type distributions, which are discussed at the end of this chapter. Just as with distributions, generalized functions are continuous linear mappings on certain testingfunction spaces. Thus, our first objective is to develop a general formulation for the needed testing-function spaces. Throughout the rest of this chapter p, v, i , , j , f, and p denote nonnegative integers in R , whereas k = { k v } t = lis a nonnegative integer in R". As before, m = {m,}:=1, where each m, is either a nonnegative integer or 03. A sequence {Kj}T= will be called a nested closed cover of R" if each K j is a closed subset of R", K j c K j + l for every j , K j = R", and each compact subset of R" is contained in one of the K j . Let {Kj}j"=oand {Ip}Pm,o be two nested closed covers of R". For each j and 1, let there be given a continuous function t j ,f(t) from R" into R such that t j ;f(t) > 0 for every t E R". Also, assume that t j ,f(t) 2 tj+l,f ( t )for all j , f, and t. For every j and p , we define the functional pi, on suitably restricted functions 4(t)from R" into A by

uj

,

Pj,p(4)

4

max

O
SUP I I t j , d t ) 4 ( k ) ( t ) I I A 7

?€I,

(1)

where it is understood that the maximum is also taken over v = 1 , . . . , n. Clearly, P j , p ( 4 ) P j , p+1(4). For each j , we define $,"(A) as the linear space of all functions 4 from R" into A such that c $ ( ~ ) is continuous whenever 0 Ik m, supp 4 c K j , and pj,,(+) c 03. In this case, each p i , , is a seminorm on .Yj"(A) and, if 4 is not identically zero, then pi, ,(4) > 0 for at least one of the pi, ,. We assign to Y j " ( A ) the topology generated by { p i , ,},"=o and obtain thereby a metrizable locally convex space. That Y j m ( A )is complete and therefore a FrCchet space can be shown by using the result asserted in Problem 1.6-3. Clearly, the function 4 H 4(k),where k I m, is a continuous linear mapping of Y j m ( A ) into Y y - k ( A ) . Both Y j " ( A ) and its topology remain unchanged if the p j , , are altered under any one of the following three situations. (i) If K j is bounded, all the I, can be set equal to R" and t i ,f(t) can be set equal to 1 for all t and 1. In this case, Y j " ( A ) = 9:&4). (ii) If any one of the I, is equal to R", all the I, can be set equal to R". (iii) If every I, is bounded, {I,} can be replaced by any other nested closed

3.6.

THE

p-TYPETESTING FUNCTION SPACES

cover of R" consisting exclusively of bounded sets, and, in addition, can be set equal to 1 for all t and 1.

65

t j ,,(t)

Since t i ,, ( t ) 2 tj+l,, ( t ) for allj, I, and t , it follows that p j , p ( 4 )2 ~ ~ + , , ~ ( 4 ) for every 4 E 9 j m ( A ) . Furthermore, K j c K j + ] . Consequently, Yj"'(A)c 9[i"+ , ( A ) for allj, and the topology of 9 j " ( A ) is stronger than that induced on it by 9[i"+ ] ( A ) . We define $"(A) 4 9 j m ( A ) .This is a linear space, and we equip it with the inductive-limit topology (Appendix El). We shall call any locally convex space of functions having this form a p-type testing-function space, and 9"'(A) will always denote such a space. Clearly, a sufficient condition for 9"' ( A ) to be strict as an inductive-limit space (Appendix E3) is that t j , ,= < j + l , for every j and 1. If, in addition, K j = R" for at least one j , then Y m ( A )= 9 j " ' ( A ) ,in which case P ( A ) will be called degenerate. As usual, we set 9 r m 1 ( A ) = 9 ( A ) , . P ( C ) = 9"', and 9 [ " l ( C ) = 9.This notational convention is followed for every one of the special cases described below. For subsequent use in defining a topology for [ 9 " ( A ) ; B ] , we single out a certain collection 6 of subsets of 9 " ' ( A ) . The members of 6 will be called 6-sets. A subset Q of 9 " ( A ) is said to be an 6-set if i2 is contained in some Y j ( A ) and is a bounded set therein. Thus, every 6-set is a bounded set in 9 " ( A ) , but the converse is not true in general. The converse (namely, the bounded sets in 9 " ( A ) are 6-sets) is true whenever 9"'(A) is a strict inductivelimit space with the closure property (see Appendix E4). We shall now list a number of special cases of 9 " ( A ) . In the following, a locally convex space Y of A-valued functions on R" will be called normal if 9 ( A ) is a dense subspace of Y and the canonical injection of 9 ( A ) into Y is continuous.

uT=

,

1. 9 " ( A ) . This occurs when all the K j are bounded sets. In this case, we can replace { K j } by any other nested closed cover of R" consisting of bounded sets. By virtue of Lemma 3.2-3, B"(A) is normal. For fixed m, 9 " ( A ) is the smallest p-type testing-function space in the following sense. For any other p-type testingfunction space 9"'(A), we have that 9"'(A) c 9"'(A). Moreover, the canonical injection of 9 " ' ( A ) into 9 " ( A ) is continuous. 11. &"'(A). We obtain this case when every K j is R" and every I,, is a bounded set. As was indicated before, we can now set t j ,, ( t ) = 1 for allj, I, and t . Hence, &"'(A)is degenerate, and we have pj, p ( 4 ) = pp(4> =

max

SUP

O ~ k , s m i n ( p ,m,) tsl,

II 4'k'(t>IIA

&"(A) is a Frichet space and is normal. For fixed m, &"(A) is the largest of the p-type testing-function spaces because 9 " ' ( A )c &'"(A) for any other space 9 " ( A ) . Indeed, every A-valued function 4 on R" such that 4(k)is

3.

66

BANACH-SPACE-VALUED DISTRIBUTIONS

continuous whenever 0 5 k 5 m is a member of €"(A). Thus, l l ~ ( k ) ( t ) /has l no restriction on its growth as I t ( co. Furthermore, the canonical injection of 9 m ( A )into &"(A) is continuous. I l l . Y m ( A ) . Now, K j = R", I , = R", and -+

tj, ~

t =) (1

+ I t I '1'

for everyj, q, I, and t . Thus, Y m ( A )is degenerate, and Pj,p ( 4 ) = ~

SUPII(~ +

max

p ( 4 = )

Oqkv4min(p,mv)teR"

I I 2>p4'k'(t>IIA.

Here, too, Y m ( A )is Frechet and normal. The members of Y P [ " ' ( A )= Y ( A ) are called A-valued testing functions of rapid descent. Now, l\+(k)(t)l/ tends to zero faster than any negative power of [ t I as 1 t I 00. IV. Y Z d ( A ) . Let c = {c,}:, E R" and d = {d,,}:= E R". Set

n n

Kc,d ( l ) =

Kc,. dy(fv)?

v= 1

where

Also, let K j = R", I , = R", and P j ,p ( 4 )

t i ,, ( t ) = K c , , ( ? ) for allj, q, 1, and t . So, max

= Pc, d, p ( 4 ) =

Ock.crnin(p,m,)

"P 11 ' c ,

d(t>4'k'(t)ll A

'

feR"

The resulting p-type testing-function space 2 ' : d ( A ) is degenerate and therefore Frtchet. It is not normal because g m ( A )is not dense in Y E d ( A ) .This space and the next one arise naturally in the study of the generalized Laplace transformation (Zemanian, 1968a, Chapter 3). The members of these spaces have exponential bounds on their behavior as 1 t I -+ co. where each w, is either a real number V. 9 " ( w , z ; A ) . Let w = { w,,}:' or - co, and let z = {z,};=1, where each z,, is either a real number or 00. Let {cj}T= and {dj}T= be two sequences in R" with the following properties: cj+l < c j and d j + l > dj for every j . Also, upon denoting the components of c j by c j ,,and of dj by d j ,,, where v = 1, . . . , n, assume that, for each v and as j - + co, c j , ,+ w, and d j ,,-+ z,. Now, set K j = R", I, = R", and ti,, ( t ) = 2 tj+l, as is required. ~ , ~ , d , ( t )for every j , p , f , and t . Thus ti,,(?) Moreover P j ,p ( 4 ) = P c j , d,,

p(4)

=

max

S U P l l K c j , d,(t)4'k'(t)llA O < k v < m i n ( p . m,) feR"

Thus, Y m ( w ,z ; A ) is the inductive-limit space not strict. On the other hand, it is normal.

.

ujY ; , d j ( A ) . However, it is

3.7.

GENERALIZED FUNCTIONS

67

VI. 9-"'(A). In this and the next special case, n = 1, so that we will be dealing with functions on the real line R. Also, k is now a nonnegative integer in R , and m is either a nonnegative integer in R or 00. We set K j = ( - co,j ] ,I , = [ -p, a),and ti,I ( t ) = 1 for allj, p , 1, and t . Thus, the functions 4 in 9 - " ( A ) have their supports bounded on the right, and Pj,p(4)

=

max ~ ~ P I I ~ ' ~ ' ( ~ ) I I A *

osksp

rErp

However, l14(k'(t)llhas no restriction on its growth as t + - 00. The space 9-'"(A) is a normal space as well as a strict inductive-limit space having the closure property. VII. 9+"'(A). Here again, n = 1. Also k and m are as in the preceding case. We set K j = [ - j , a), I , = (- 00, p ] , and t,, I ( t ) = 1 for all j , p , 1, and t . The functions in 9 + " ' ( A )have their supports bounded on the left, and

D+"'(A)is both normal and strict and has the closure property. Problem 3.6-1. If I , 3 gi,then p i , is a norm on $,(.A) for every p 2 q. Conversely, if pi, is a norm on $,(A), then 1, 2 g j .Prove these assertions.

,

Problem 3.6-2. Prove that Ij"(A) is complete. Problem 3.6-3. Show that the differentiation operator Dk is a continuous linear mapping of P ' + k ( A )into . F ( A ) . Problem 3.6-4. Show that &"(A), Y " ( A ) , Y"(w, z; A), .9-"'(A), and 9 + " ' ( A )are all normal. Also, show that the shifting operator or is an automorphism on each of these spaces.

3.7. GENERALIZED FUNCTIONS

Given any p-type testing-function space $"'(A), a continuous linear mapping of F ( A ) into B is said to be an [ A ; B]-valuedgeneralizedfunctionon R". As we done with distributions, we will subsequently identify [$"; [ A ; B]] with [$"'(A); B], and thereby [P'; [C; B]] with the space [P; B] of B-valued generalizedfunctions on R".Since $"'(A) contains 9 ( A )and induces a topology on 9 ( A ) weaker than that of 9 ( A ) , the restriction of any f E [$"'(A); B] to 9 ( A ) is a member of [ 9 ( A ) ;B] (Le., is an [ A ; B]-valued distribution on R"). If P ( A ) happens to be normal, [$"'(A); B] can be treated as a subspace of @ ( A ) ; B] by identifying each f~ [$'"(A); B] with its restriction to .9(A).

68

3.

BANACH-SPACE-VALUED DISTRIBUTIONS

This is because the said restriction determines f uniquely on all of $"'(A) by virtue of the density of 9 ( A ) in $"'(A). The topology we will usually employ for [$'"(A); BJ is the topology of uniform convergence on the 6 - s e t s in $"'(A), which we will simply call the 6-topology. This is the topology generated by the collection {ye}ess of seminorms defined by

ydf)

suPll
4EQ)

where/€ [$'"(A); B ] and cD is an arbitrary 6-set. To indicate that this is' the topology that is being employed, we will use the notation [$'"(A); B]". However, when $"'(A) is either degenerate or more generally a strict inductive limit having the closure property, then the 6-sets are precisely the bounded sets in $'"(A), so that the 6-topology is the boundedtopology. In this case, we drop the superscript s on the notation [$"'(A); B]. This is the situation for each of the spaces [9'"(A);B], [&'"(A); B], [Y"'(A); B], [LZ':,(A); B], [ 9 - " ' ( A ) ;B ] ,and [9+"'(A);B]. Still another topology is the pointwise topology generated by {yo}, where 4 traverses 9'"(A) and

rdf) I1(f,4) II When using this topology, we employ the notation [$'"(A); B]".

Lemma 3.7-1. Let F be a bounded set in [$"'(A); 81".Then, corresponding to each j , there exists a nonnegative integer p E R and a constant M > 0 such that s ~ ~ l l (4)lliJ f , 5 MPj,p(4)

(1)

JEF

for all 4 E Yj'"(A).M andp depend in general on Fandj.

PROOF.Let Fj be the set of restrictions to .Yj'"(A) of all f E F. Then, Fj is a bounded set in [Yj'"(A);B]". Since Yj"'(A) is a FrCchet space, we may invoke Appendix D9 to conclude that Fj is equicontinuous on f j ' " ( A ) , or, in other words, that (1) is satisfied. 0 Clearly, if F consists of a single element, then F satisfies the hypothesis of the lemma. Another special case arises when F is a bounded set under the 6-topology; for, in this case, F is also bounded under the pointwise topology because the pointwise topology is weaker than the 6-topology. Generalized differentiation Dk is defined on any generalized function f E [9"'(A);B] exactly as it is on any distribution, namely

(0% 4) 4 (- l)lkl(xD"),

9 E $"'+k(A).

By means of a proof exactly like the one for Theorem 3.3-1, we have the following result.

3.7.

GENERALIZED FUNCTIONS

69

Theorem 3.7-1. Dk is a continuous linear mapping of [ 9 " ( A ) ; B]' into [.P""k(A);B]', as wellasof [#"'(A); B]" into [F""k(A); El". For future reference, we now list a number of spaces consisting of linear combinations of certain elements.

9"0A . This is the span of all elements of the form 4a A a+, where and 4a denotes the mapping t ~ + ( t ) u .Thus, 9"0A is a subspace of Y"'(A). 11. [9"; C ] and a E A , g a ag is defined by the C ] 0A . If g E 19"'; equation I.

4 E 9,a E A ,

(ga,

4 ) A (9,

$>a,

4 E 9".

(2)

I t follows that ga E [9"; A ] . [9"; C ] 0A denotes the span of all elements of the form ga and is therefore a subspace of [Y"';A ] . 111. [ 9 " ' ; [ A ;B ] ] 0 A . IfgE[Y"'; [ A ; El] and a E A , we definega by Equation (2) again. Now, ga E [9"; El. [P;[ A ; El] 0 A denotes the span of all elements of the form ga and is therefore a subspace of [Y";El. Our next objective is to relate [ Y ' ( A ) ; B ] and [9"; [ A ; B ] ] . Through precisely the same proof as that of Lemma 3.3-2, we can establish the following lemma.

Lemma 3.7-2. Eiiery f E [ 9 " ( A ) ;B ] uniquely dejines a g E [Y"; [ A ; El] by means of the equation

(a $)a

4 (f, $a>,

$ E Y", a E A.

(3)

Theorem 3.7-2. rf' 9"' and P ( A ) are normal spaces, theii there exists a [ A ; El] dejinedby (3). bijectionfrom [9"'(A);B ] onto [P; PROOF.The steps of this proof are illustrated in Figure 3.7-1. Given f E [ 9 " ( A ) ; B ] , we may replace (3) by (f, 4a> = ( 9 , 4 > a ,

(4)

where 4 is restricted to 9, and still obtain a definition for g E [9"; [ A ; B]]. This is because 9 is dense in 9"' so that we need merely specify g on 9 in order to determine g on all of 9'"Let . F4:f-g denote this mapping. Now, let g E [9"; [ A ; El] and let 8 be the restriction of g to 9. Because of the normality of 9", the mapping F,: g H @is a bijection of [9"; [A; B]] onto a subspace U of [9; [ A ; B ] ] . Moreover, we may write (g, 4)a

= (89

$>a,

4 E 9,a E A .

(5)

70

3.

BANACH-SPACE-VALUED DISTRIBUTIONS

By Theorem 3.5-1, the equation

(G, 4 h

(f,4a>,

where 4 E 9,a E A , 4 E [9; [ A ; R ] ] , and f~ [ 9 ( A ) ;B], defines a bijection F6: Qwf of [ 9 ; [ A ; B ] ] onto @ ( A ) ; B ] . Thus, the subspace I/ is mapped by Fti in a one-to-one fashion onto a subspace V of [ 9 ( A ) ;B]. O n the other hand, let us denote the restriction of any f E [ Y ' ( A ) ; B] to 9 ( A ) by f. Then, the mapping F7: f w f is a linear bijection of [ 9 " ( A ) ;B] onto a subspace W of [ 9 ( A ) ;B] because of the normality of Y"(A). Moreover, by Lemma 3.5-1, the span of the set of all elements of the form +a is dense in 9 ( A ) . Therefore, the bijection F7 is determined by

( A 4a) 4i

.

(7)

Upon combining (4)-(7), we see that the composite mapping Fb F5 F4 is equal to F 7 . Therefore, the subspaces V and W must coincide, and, since F7 is a bijection of [ 9 " ( A ) ; B] onto W , so too must be Fti F, F 4 . But we have already noted that F, and Fti are bijections. Consequently, F4 must be a bijection. 0

3.7.

GENERALIZED FUNCTIONS

71

Because of Theorem 3.7-2, we will customarily denote f and g by the same symbol and will replace (3) by ( f , *>a = ("6 *a>.

(8) We end this section by noting certain support conditions possessed by the members of [&"'(A); B] and [ 9 + " ' ( A ) ;B ] . These members are distributions since both &"'(A) and 9+"'(A)are normal.

Theorem 3.7-3. f E [&"'(A); B] ifand only iff compact set.

E

[9"'(A);B] and supp f is a

PROOF.Given any f E [&"'(A);B ] , there exists a neighborhood A of 0 in €"'(A) such that II(f, 4)llB2 1 for all 4 E A. Upon referring to Case I1 of the preceding section, we see that A contains a set SZ consisting of all 4 E &"'(A) such that max supI14'k)(t)IlAIE , 0 s k y $ m i n ( p , m,) f

EI,

where I , is a compact set. This implies that supp f c I , . Indeed, iff is not equal to the zero distribution on R"\I, , then we can find a.O E 9 ( A ) such that supp 0 c R"\I, and ( f , 0) + 0. But then, M0 E R for all real numbers M , and I I ( f , MO)I/ can be made larger than 1 by choosing M appropriately. This is a contradiction. Conversely, iff E [ W ( A ) ; B ] and supp f is a compact set, then, by virtue of the paragraph just before Example 3.3-1, f has a unique extension onto any 4 E €"'(A) defined by ( f , 4 ) = ( f , A4),where A E 9 is equal to 1 on a neighborhood of suppf. This extension is clearly linear and is continuous because the convergence of {&,,} to 0 in &"'(A) implies the convergence of {h$"} to 0 in 9 " ' ( A ) . 0 A similar argument establishes the following result.

Theorem 3.7-4. . f [ ~ 9 - " ( A ) ; B] (or f E [ 9 + " ' ( A ) ;B ] ) fi and only sf f E [ W ( A ) ;B] and suppf is bounded on the left (or respectively on the right). Problem 3.7-1. Show that, when 9"'(A) is normal, the restriction of any , f ~[Y"'(A);B ] to 9 ( A )uniquely determinesfon all of 9"(A).

Problem 3.7-2. Prove Theorem 3.7-4. Problem 3.7-3. Let .P"'(A) be any one of the spaces &"'(A), 9'"'(A), z ; A ) , &"'(A), and 9+"'(A). Show that the shifting operator is an automorphism on [9"'(A);B]. ~""(MJ,

72

3.

BANACH-SPACE-VALUED DISTRIBUTIONS

Problem 3.7-4. Every continuous A-valued function is a regular A-valued distribution (see Example 3.3-2). The canonical injections of &''(A) into [go; A ] and of 9 ' ( A ) into [ g o ;A ] are continuous. Verify these assertions. Problem 3.7-5. Show that 9 ' ( A ) is a subspace of [Yo; A ] for every p-type testing-function space 9'. Problem 3.7-6. Let % be a continuous linear mapping of 9 ( A ) into [9;B]. Define 5Y.l as a mapping on 9 by

('JJzf, $ ) a

(%(fa), 4 ) ,

(9)

where,f E 9, 4 E 9, and a E A. Show that llJl is a continuous linear mapping of 9 into [9;[A; B]].

uj

uj

Problem 3.7-7. Let Y" = .fj"and f' = f ; be p-type testingfunction spaces consisting of complex-valued functions. Let % be a continuous linear mapping of [Ym;A]" into [f'; B]'. Define a mapping 91 on [Y"; C ] by (9), where now f E [Y"; C ] , 4 E y ,and a E A. Show that 9.Ji is a continuous linear mapping of [Y";C]" into [f'; [A; B]]". Also, show that this result is again valid when the'G-topologies of [Y"; A ] , [ f ' ;B ] , [Y"; C ] , and [ f ' ; [A; B ] ]are replaced by the pointwise topologies. 3.8. L,-TYPE TESTING FUNCTIONS AND DISTRIBUTIONS

The &-type testing functions do not fit the general formulation for the p-type testing functions. They are instead defined as follows. Let p E R be fixed with 1 I p < co. The space QLP(A)is the linear space of all smooth A-valued functions 4 on R" such that, for each integer k E R" with k 2 0,

We set g L P ( C = ) g L PThe . members of g L P ( A are ) said to be L,-type testing functions. Minkowski's inequality shows that y p , k is a seminorm on gLP(A). We assign to gL,(A)the topology generated by {y,, k } k > O . This separates the space because, if y,, ,,(4) = 0, it follows from the nonnegativity and continuity of III$(.)II that 4(t) = 0 for every t. Thus, 9,,(A) is a metrizable space. The shifting operator 0,: 4 ( t ) w $ ( t- r ) is an automorphism on gLP(A), and differentiation is a continuous linear mapping of QLP(A)into 9,,(A). The space gLP(A) is a subspace of the space L,(A) of all (equivalence classes of) A-valued Bochner-integrable functions with respect to Lebesgue measure on the Bore1 subsets of R" (see Appendix G19). Moreover, the canonical injection of 9&4) into L,(A) is continuous.

3.8. L p - TESTING ~ ~ FUNCTIONS ~ ~

A N D DISTRIBUTIONS

73

In turn, &(A) becomes a subspace of [9; A ] when everyf E L J A ) is identified with a mapping g of 9 into A through the equation

Indeed, g is clearly linear on 9. Moreover, for p > 1 and q = p / ( p - l ) , we have from Holder's inequality (Appendix G20) that

for all

4 E Q K , and hence g E [Q; A ] . When p

=

1 , we may write

for all q5 E QK and thereby conclude once again that g E [9;A ] . These estiA ] is continmates also show that the canonical injection of &(A) into [9; uous. We now develop some inequalities that we shall subsequently need. Let 1 + denote the n-dimensional unit-stepfunction defined by I+(?) = l + ( t l ) ... l + ( t n ) , where in the right-hand side 1 + is the unit-step function on the real line (see Section 3.4). As usual, D[ll 4 dl a,, denotes the differential operator of order [l]. Also, let 4 E €["(A) and let A E 9 be such that A(t) = 1 for It I < 1 and A(?) =,0 for I t I > 2. For any fixed t E R", we may write

A(?

- x ) ~ ( x )= ( - 1)"

Upon setting f

=x

JR" 1 +(T)D:''[A(~- x +

T)&X

- T)] dT.

and then estimating the result, we get

where M is a constant and E = { T : I t - T I < 2). More generally, if k is a nonnegative integer in R" and if q5 E gk+[ll(A), then

Also, upon applying Holder's inequality (Appendix G20), we obtain, for I
l14(k)(~)ll sN

2

O c r s [ l l [jEllq5(k+r)(T)Iv

where N is still another constant.

dr]

(2)

3.

74

BANACH-SPACE-VALUED DISTRIBUTIONS

Lemma 3.8-1. gLP(A) is a normal space. PROOF. Clearly, 9 ( A ) is a subspace of BL,(A)and the canonical injection of 9 ( A ) into QL,(A) is continuous. To show the density of Q ( A ) in BLp(A), choose arbitrarily a 4 E BLp(A). Also, let A E 9 be as before and set A j ( t ) = A(t/j) for j = 1, 2, . . . . Then, 4Aj E g ( A ) . Moreover, for any k, we may write Yp,

k(4

- 4Aj)= Y p . O{Dk[4(1 - A j ) l }

where M is a positive constant not depending on j . The integral in the last expression tends to zero as j - t co. Therefore, 4Aj -+ 4 in g L P ( A ) ,which proves the density of 9 ( A )in g L P ( A ) . 0 A simpler version of the preceding proof shows that L,(A) is also a normal space.

Lemma 3.8-2. Assume that p , q E R satisfy 1 5 q < p < co. Then, gLqis a , the canonical injection of g L q ( A )into 9 L p ( A ) is dense subspace of g L P ( A ) and continuous. PROOF. Let 4 E g L , ( A ) . It follows from (1) that l14(k)(t)ll-to as I tI + co for every k . That is, the set { t : ~ l ~ ( k ) 2 ( t I) }~ is ~ bounded. On the complement 2 l14(k)(t)llp. Therefore, 4 E g L P ( A ) . of that set, I14(k)(f)114 Since 9 ( A ) c BL,(A)c g L , ( A ) and since 9 ( A ) is dense in g L P ( A ) ,it follows that g L , ( A ) is dense in g L P ( A ) . Finally, let the sequence { $ j } tend to zero in g L 4 ( A ) We . see from ( 2 ) that, for each k , {I14y)(t)ll}jtends to zero uniformly for all t E R". Therefore, there exists an integer j o such that, for a l l j > j o ,

This implies that the lemma. 0

{ 4 j }tends

to zero in g L , ( A ) and completes the proof of

As usual, [aLP(A); B] is the linear space of all continuous linear mappings of g L P ( A into ) B. Since g L , ( A ) is a normal space, it follows that [QLP(A);B] can be identified with a subspace of [ . 9 ( A ) ;B] by identifying eachfe [ g L , ( A ) ;

3.8. L,-TYPETESTING

FUNCTIONS AND DISTRIBUTIONS

75

B ] with its restriction to g ( A ) . The members of [ g L P ( A ) ;B] are called [ A ; B]-valued L,-type distributions. On the other hand, the members of [gL,;B] are said to be B-valued since [ C ; B ] can be identified with B. As always, [ g L p ( A ) ;B] is understood to have the bounded topology unless something else is indicated. The shifting operator ozand generalized differentiation Dk are defined on [BLp(A); B] in the usual way for distributions (see Section 3.3). cr, is an automorphism on [ 9 L p ( A ) ;B], and Dkis a continuous linear mapping of [gL,(A); B] into [ g L P ( A ) B ; ]. An immediate consequence of Lemma 3.8-2 is the following one.

Lemma 3.8-3. Assume again that p , q E R satisfy 1 I q < p < co. Then, [gL,(A); B] is a subspace of [ g L 4 ( A ) B]. ; Furthermore, the canonical injection of the former space into the latter one is continuous, and the same is true when both spaces possess the pointwise topologies. Through the same proof as that of Theorem 3.7-2, we obtain the following result.

Theorem 3.8-1. There exists a bijectionfrom [ g L P ( A )B; ] onto [QL,; [A; B ] ] defined by the equation ( 9 , *>a = (f, *a>,

(3)

where $ E g L Pa,E A , g E [gL,;[ A ;Bl], and f E [ g L P ( A )Bl. ; Because of this bijection, we will usually denote both f and g in (3) by the same symbol.

Chapter 4

Kernel Operators

4.1. INTRODUCTION

This chapter starts our discussion of a realizability theory for continuous linear systems, the next section being devoted to a consideration of various types of systems that generate the operators discussed in this book. The subsequent sections are devoted to our most general class of operators, namely to operators that map QR3(A)into 19,";B ] linearly and continuously. [QR.(A) denotes the space Q ( A ) with the additional specification that its members are defined on R". As usual, 9," 9- gRn(C).]The paramount result in this chapter is the kernel theorem as stated by Theorem 4.4-1. This is an extension to a Banach-space setting of Schwartz's kernel theorem. It provides a kernel representation for the operators at hand, which is the subject of Section 4.5. How causality affects the kernel representation is discussed in Section 4.6, but the implications of other physically motivated assumptions such as time invariance and passivity are reserved for subsequent chapters. 76

4.2.

SYSTEMS AND OPERATORS

77

4.2. SYSTEMS AND OPERATORS

Let us start by presenting a physical example of a system whose signals take their instantaneous values in a Hilbert space. This example employs certain results concerning electromagnetic waves and is given only for illustrative purposes. It is not essential to any succeeding discussion. Example 1.2-1. Consider an electromagnetic cavity resonator that is being excited through a rectangular waveguide as indicated in Figure 4.2-1. The unit vectors for the Cartesian coordinate system {x, y, z } are I,, I,, and 1,; 1, is directed along the axis of the waveguide, as shown. d will denote the cross-sectional surface cut by a fixed transverse plane at some point along the waveguide. Let E be the complex electric field intensity vector and F the complex magnetic field intensity vector at some arbitrary point of d.Thus, E and F are complex vectors depending on {x, y , t } , where t denotes time.

Figure 4.2-1

We let E, and Fa be the projections of E and F onto d,and therefore Ea and Fa are complex vectors lying in the plane of d.The instantaneous net complex power passing through d toward the resonator is P(t) =

j ( E x F ) . 1, JB

da =

1(E,

x Fa) * 1, da,

d

where the bar denotes the complex conjugate, x the cross product, . the dot product, and du the incremental area on d.The corresponding real power is Re P ( t ) .

4.

78

KERNEL OPERATORS

Now, assume that Ea and Fa are quadratically integrable on d ;that is,

According to Jones (1964, p. 246), there exist real basis fields e j andfi, where j = I , 2, . . . , having the following properties. e j andfi are real vectors lying in the plane of d and depending only on { x , y } ; they do not depend on t . Moreover, fj = 1, x e j , {ej)j", is the orthonormal basis of some separable complex Hilbert space H , and

where d j 9 denotes the Kronecker delta. Finally, Ea and Fa have the unique expansions m

03

Ea= c u j e j ,

Fa= c u j f j ,

j= 1

j= 1

where v j and u j are complex-valued functions of t but do not depend on (x, y } . As a consequence of these properties, we have the identity S,(.j

fk)

*

=

I

(fj

d

*fk)

da

= 8j,k

and the equation

The point of this example is the following observation. We may view the system of Figure 4.2-1 as generating an operator '3 that maps the quantity u j e j . Both u and u are functions u Ea = zijej into the the quantity u of t taking their values in the complex Hilbert space H. Moreover, the instantaneous net complex power P ( t ) into the system is given by

1

p ( t ) = (u(t>,v(t)) = 2 uj(t)vjo,

where (., .) is the inner product for H . Actually, u is a fictitious quantity, which is uniquely related to Fa. Indeed, given any Fa = u j f , , one need merely replace.6 by e j to obtain u. Upon borrowing some concepts from electrical network theory, we can consider u to be an H-valued voltage signal and u an H-valued current signal on the system of Figure 4.2-1. Thus, 91 can be taken to be an admittance operator. Were H n-dimensional Euclidean space, our system would be called an n-port by network theorists. In analogy to this, we shall call the present system a Hilbert port.

1

4.2.

SYSTEMS AND OPERATORS

79

We can view our system in a somewhat different way to obtain a closer analogy to the electrical n-port. We exploit the isomorphism between H and the Hilbert space e,; that is, we identify u with its sequence {uj} of Fourier coefficients and u with {uj}.The coefficient u j is considered to be the voltage signal at thejth port of an electrical system, and u i the current at that port, as indicated in Figure 4.2-2. The polarity of uj andthe direction of uj are so chosen that uj(t)vj(t) denotes the instantaneous power entering the jth port. Thus, we have arrived at an n-port, where now n = co. One might call this an co-port (Zemanian, 1972, Section 8). This ends our discussion of Example 4.2-1. 0

co-port

I I

I Figure 4.2-2

It is only when considerations of power flow arise that an inner product is needed for the space in which the signals at hand take their values. In this book, this will occur only when we impose the assumption of passivity, which states in effect that the system does not contain energy sources that can transmit energy to the exterior of the system. Thus, in the absence of passivity, we can and shall assume that the signals take their values in Banach spaces. In general, therefore, our analyses will involve signals that are Banachspace-valued functions or distributions. We shall refer to a system having such signals as a Banach system. It should also be pointed out that it is usually

80

4. KERNEL OPERATORS

to transient phenomena that the realizability theories of this book are applied. This means that time is the independent variable for the signals, which are therefore functions or distributions on the real line R. However, there do occur physical phenomena involving signals on R”that are amenable to some of the subsequent theories. An example of this is the optical system discussed by Meidan (1970). For this reason, we shall allow, at least initially, signals defined on R”. A Banach system may have many different Banach spaces associated with it. For example, the signal u representing some physical variable at one location x within the system may be an A-valued distribution, whereas the signal u for another physical variable at a different location y may be a B-valued distribution. Moreover, the system defines the relation %: U H U . % need not be an operator; that is, more than one u may be assigned to some particular u. (An example of this is the ideal transformer of electrical network theory when u is taken as the current vector and u as the voltage vector.) Howeuer, a basic assumption imposed throughout this book is that every relation with which we shall be concerned is truly an operator. Furthermore, a Banach system may define many different relations depending on the choices of the locations x and y and the physical variables. The term ‘‘ Banach system ” refers to the entire system and not to any particular relation generated by it. Moreover, it is possible for certain operators generated by a given Banach system to exhibit various properties such as linearity, time invariance, and passivity, while other operators generated by the same system do not. For this reason, the postulates in our subsequent realizability theory will be imposed on particular operators and not on the system as a whole. Indeed, it is such operators and not entire systems that comprise the main concern of this book. Let us now define the concept of a “ Hilbert port.” Assume that in a given Banach system we have singled out two physical variables u and u that are complementary in the following sense: When both u and u are ordinary functions taking their values in a (not necessarily separable) complex Hilbert space H, the inner product (u(t), u(t)) represents the net complex power entering the Banach system at the instant t . Then, the Banach system with these two variables so singled out is called a Hilbert port. We shall borrow some terminology from electrical network theory by referring to the relation %: U H U , when it is truly an operator, as the admittance operator ofthe Hilbertport, and to the relation 2B:$(u + U)H+(U - u), when it too is an operator, as the scattering operator of the Hilbert port. This agrees with the usual terminology for electrical n-ports when u is identified with the voltage vector and u with the current vector. For the scattering operator, certain normalizations of u and u are also implied by this (Carlin and Giordano, 1964, p. 225). It is the admittance and scattering operators with which we shall be concerned when the passivity hypothesis is imposed.

4.3. THE SPACE 2 = g(Y)

81

Still another operator we could consider is the impedance operator 3 :U H U , where again u is current and v is voltage. However, everything we shall say about ' i l lcan be applied equally well to 3 by interchanging the roles of u and u.

4.3.

THE SPACE 2 = 9 ( Y )

Schwartz's kernel theorem (Schwartz, 1957, p. 93) characterizes separately x g R Sinto C in terms of complexcontinuous bilinear mappings of gRn valued distributions on R" x R". There now exist a number of alternative proofs for it (Bogdanowicz, 1961 ; Ehrenpreis, 1956; Gask, 1960; Gelfand and Vilenkin, 1964, Section 1.3). For our purposes, we need an extension of this theorem to separately continuous bilinear mappings of g R nx gR.(A) into B. Actually, Bogdanowicz's proof establishes the kernel theorem for mappings of gRnx 9 R s ( A ) into C, and with some obvious modifications, we can replace C by B. His proof is the subject of this and the next section. The present section is devoted to a generalization of the space g R , ( A ) resulting from the replacement of A by a more general type of space V . In the following, we let Y be the strict inductive limit of a sequence {Yj}y=of FrCchet spaces. Since every FrCchet space is separated and complete, we can conclude that Y possesses the closure property (Appendix E4). For each j , we let Z j denote a sequence { [ j , y } ~ = oof seminorms that generates the topology of Y j. We can always choose the multinorm Z j such I(,,z I . this we do. As a consequence, a base of that [ j , o I neighborhoods of 0 in Y jconsists of all sets of the form

cj,,

* a ;

{v

E Y j :5 j , q ( ~ ) < E } ,

where E E R + and q are arbitrary. Now, let { K j ) y = l be a sequence of compact intervals in R" such that K j c Rj+l for every j and K j = R". We let 2 4 gRn(Y) denote the linear space of all smooth Y-valued functions on R" having compact supports. It follows from Theorem 1.3-1 that, for any h E Z , h(R") is a bounded subset of Y . Consequently, according to Appendix E4, h(R") c V j for some j depending on h. We now let Z j A gK,(Yj) be the linear space of all h E 2 such that h(R") c V j and supp h c K j . Thus, Z j c X j + , for every j , and Z=UZj. Fix j and consider Z j . For any h E Z j and any nonnegative integer p E R", h ( p ) is a continuous function from R" into V and its range is contained in " Y j . Since the topology of Y j is identical to the topology induced on "Yj by "Y, h ( p ) is also continuous from R" into V j .This means that, for any

u

82

4. KERNEL OPERATORS

given 5 E Z j and p , we can define a finite-valued functional xp, on X i by means of x p , c ( h )Li sup [[h'P'(t)], h e S j . 1E

R"

Each xp, is a seminorm on S j .We equip i?Vj with the topology generated by the collection Ti4 {xp,<}of all such seminorms. This makes makes Z j a metrizable locally convex space. A useful and easily shown fact is the following. Let k and p be integers in R" with 0 5 k 5 p , and let w and [ be seminorms in Z j with w 5 [. Finally, let T = diam K j . Then, for all h E S j , (1)

Xk,,,(h)I T ' P - k ' x p , c ( h ) .

Lemma 4.3-1. The topology of i?V, is identical to the topology induced on xj by % j + l .

PROOF.Since the topology of Y j is the same as the topology induced on Y j by V , + every set of the form {v E Y j :[ ( u ) < E , [ E Z j } contains a set of the form { u E Y j :[ ( v ) < 2, [ E Z j +I } , and conversely. Now, consider the following neighborhood of zero.in Z j : ( h e S j : sup [[h'P'(t)]< E ,

[E Zj

f

1

.

(2)

By our first observation, the set (2) contains a neighborhood of zero in the induced topology of X j of the form

I

: sup [[h(P'(t)]< &, p E Z j + , . (3) f (h But a base of neighborhoods of zero in .?Pi consists of all intersections of finite collections of sets of the form (2), and similarly for the induced topology with respect to (3). Therefore, we can conclude that the topology of i?Vj is weaker than the topology induced on it by X i + The same kind of argument proves that the topology of i?Vj is stronger than the induced topology. 0

u

Henceforth, we assign to i?V = X j the inductive-limit topology. In view of Lemma 4.3-1, this makes S the strict inductive limit of { S j } .

Lemma 4.3-2. Given any h E S j- with j > 1, any seminorm x p , E Tj , and any E E R + , there exists an h, E S j such that xp,s(h - h,) < E and h, is the following sum of a$nite number of terms: ho Here,

+ +4 k

= 4l~1

4, E gK,and v, E V j ,where

* * *

s = 1,

~

. .. ,k .

k

.

(4)

4.3. THE SPACE 3E" = Q('%'")

83

PROOF.We can choose a function t+b E gK,such that $ ( t ) = 1 for all ~ E K ~F o- r t~, 7.E K j , s e t g ( t ) = B p + [ 1 1 h ( t ) ,

and

L=

IQ(~J)I.

SUP f,reKj

Here, p and p are nonnegative integers in R",(3is the n-dimensional binomial coefficient, and p! ( P I ! ) . . . (P"!), ( t - t)? 4 (tl . . (r, - ~ , , ) f l n .

Now, let E E R + be given. Since g has a compact support, g is uniformly continuous on R", and therefore there exists an E R , such that

whenever 1 t - 7 1 < cl. Next, we can find a finite collection { Ri}r= of open sets such that R, 3 K j - 1 , Ri c K, , and diam ni < E I for every i. Moreover, we can find a finite collection (.ti]E1 of smooth nonnegative functions on R" such that supp l i c R i for every i, l i ( t ) = 1 for all t E K j - , , and A,(t) < 1 for all t E R". [That all this can be done is shown, for example, in the volume by Zemanian (1965, Section 1.8).] For each i, choose a t i E R i and set

0

c

c

m

go(t) 4

Then, for every t

E R", we

C Mt>g(ti)* i= 1

may write

4 " d t ) - go(t)I = C{C Ai(t)Mt) - g(ti)I)

because of (5). Let us define the integration operator I p + [ l 1on any

x E &' by

84

4.

KERNEL OPERATORS

h, is of the form asserted in the lemma. Since h = I P + [ l 1 gE 9K,-l(V) and $ ( t ) = 1 for t E K j - l , we also have that DPh(t)= Dp[$(t)ZP+[llg].

Hence, by differentiating under the integral sign, we get

Thus,

where the last inequality follows from (6). 0 Example 4.3-1. We end this section by describing a special case of .%' obtained by setting Y = g R S ( A )and Y j = gL,(A), where j = 1,2, . .. and the L, are compact intervals in, R" such that L, c Lj+l for every j and L, = R". In this case, one multinorm Z = {~,}~=,will serve for every gL,(A) when we set

u

[,(u)A max

SUP

O s k ~ I q rl e R *

IIdk)(~)IIA, u E g R s ( ~ ) .

(7)

We can identify &' with the strict inductive-limit space g R n x R . ( A = ) gRn+.(A) as follows. Let t E R" and x E R'. Furthermore, let 4 E g R n + . ( A )be given. Then, the mapping h : t H + ( t , .) is a continuous function from R" into g,.(A) because, for each nonnegative integer p E R", { t , X > H DxP&t, x) is a uniformly continuous function on R" x R" and has a compact support therein. Moreover, (Dkh)(t)= D,"r$(t, .). So, the same considerations show that h is a smooth function from R" into g R S ( A ) .Clearly, supph is bounded. Thus, to each 4 E g R n + . ( Athere ) corresponds a unique h E &'. Conversely, let EX be given. We define an A-valued function 4 on R"+"by setting 4(?,x) = [h(r)](x).The function 4 has a compact support in R"+' because h has a compact support in R" and the range of h is contained in some D,,(A). That 4 is smooth can be shown in the following way. The fact that h is continuous from R" into g R , ( A )implies that, for any nonnegative integer k E R", for a fixed t E R", and as At + 0 in R", D,"#(t + At, x) tends to D,k$(r, x) uniformly for all x E R". But D,"$(r, x) is a continuous function of x for each fixed t . We can conclude therefore that D,k4(t, x) is a continuous function of { t , x} whatever be k .

4.4.

85

THE KERNEL THEOREM

Next, let di 4 d,, denote differentiation with respect to the ith component of r. By setting up the incremental definition of d i h and taking the limit, we see that ai$(t, x ) = [dih(t)](x).After proceeding exactly as in the preceding paragraph, we can conclude that D,k a,, 4 is a continuous function on Rn+S , whatever be k. Moreover, Theorem 1.6-2 implies that the order of differentiation in DXka,, 4 can be changed in any fashion without altering the result. These arguments can be applied to all the derivatives of 4, which leads to the conclusion that 4 is a smooth A-valued function on R"". Since supp is bounded, 4 E 9Rn+s(A). We have so far shown that the equation ti

h(4 = 4(t, * > (8) sets up a bijection from X onto aRn+.(A).Clearly, this bijection and its inverse are linear. On the other hand, we have from (7) that Xp,c,(h)= max SUP lI4%Wt, x>llA. Osks[ql

f E

R";

XE

R'

This shows that, given the compact intervals Kjc R" and L j c R', the bijecConsequently, ,. tion defined by (8) is an isomorphism from X j onto g K J Y Z the bijection is also an isomorphism from 2f Onto gRn+s(A). Problem 4.3-1. Prove (1). Problem 4.3-2. Show that X possesses the closure property; i.e., for each j , Z j is a closed subspace of 2fj+l. 4.4. THE KERNEL THEOREM

We continue to use the notation defined in the last section. Moreover, we can let { p l } z 0 ,where Pl(4)

max

SUP

Osk<[ll fsRn

I 4'k'(ol

3

-

be the multinorm for aKj whatever bej. Clearly, p o p 1 5 p2 5 * * . The primary result of this chapter is the following vector version of the kernel theorem.

Theorem 4.4-1. Corresponding to every separately continuous bilinear mapping 1111 of BRnx V into B there exists one and only one 2 E [ X ;B ] such that

9IV4,v) = 2(44 for all

4 E BR,,and u E -Ir.

(1)

4.

86

KERNEL OPERATORS

Note. A much easier fact to establish is the converse: Any given 2 determines a unique (YR by means of (1). PROOF. For every pair of positive integersj and i, 9 . 3 3 1 is a separately continuous bilinear mapping of kaK,+l x Y i into B because g R nand Y are inductive-limit spaces (Appendix E2). Moreover, both gK,+and Y i are Frtchet spaces, and their respective multinorms { p l } and { ( i , q } are monotonic increasing. Therefore, we can invoke Appendix F2 to conclude the following. With g K , +and , Y ifixed, there exists a p 1 and a li,,such that

II~(4,u)IlB

MPd4)Ci,q(v)

(2)

for all 4 E g K j +and all u E Y i . such that O ( t ) Now, let 0 be any nonnegative member of gRn It1 2 1 and JR"

Also, for each j , let

E, E

=0

when

O(t) dt = 1.

R be such that

0 < E , < inf{d: d

=

15 - X I ,

5 E K,,

x E R"\K,+,]

and, as j + co,c j tends monotonically to zero. Set q j ( t ) = E,"0(t/cj). Then, (qj}y=l tends to the delta functional, and supp q, is contained in the ndimensional sphere { t : I t 1 < E,}. As a result, for any 4 E g K ,the , convolution product q i * 4 is a member of 9Kj+r for every i 2j.Also, as i-, co, qi * 4 tends to 4 in 9 K j + l . Observe that r ~ q , ( - T) is a continuous mapping of K j into 9 K , + l . Also, any given h E X belongs to some Z i ,and therefore T H ~ ( T ) is a continuous mapping of K, into Y i .By virtue of ( 2 ) , T H91(qj(. - T), h ( ~ ) is ) a continuous mapping of K, into B. Hence, 9ul(qj(*- T), h ( ~ )dz )

2,(h)

(3)

KI

exists as a Riemann integral, and, in view of ( 2 ) , satisfies the following estimate:

II Tj(h)llB 5 MHO,ci, g(h> SUP

P [ [ V ( . - T)IVOI Kj.

(4)

?€Kj

This shows that the linear mapping 2,:Z i-+ Bis continuous. Since this is true for every i, 2, is continuous and linear from Z into B as well. We set h(z) = ~ ( T ) o ,where 4 E g K ju, E Y,, and j is arbitrary. Also, let i 2j. Because of the bilinearity of 9JI, we may write z i ( + u ) = IK,m(qic- TM(T),

0)

d ~ .

4.4.

87

THE KERNEL THEOREM

But T H ~ ~ (-* T ) ~ ( T ) is a continuous mapping of Ki into gK,+, and $H%N($, u) is a continuous linear mapping of .9K,+1 into B. So, by note I1 of Section 1.4,

zi(4u) = .I(J~,vie - T ) $ ( T )

dT, 0)

= .I( *? 4., (5) We have noted that q i * 4 + 4 in g K , +as, i + co. Consequently, ( 5 ) tends to .I( u)$ in B, whatever be 4 E g K ,and u E Y , . Since j can be any positive integer here, $9

lim Zi(4u) = .I(q5, u),

q5

E 9,

u EY. 0

i-m

We now state a lemma but postpone its proof.

Lemma 4.4-1. Given any j , there exists a constant M I > 0 and a seminorm x p , in the multinorm Tifor Xisuch that ll2i(h)llB 5 MIxp,&h) (7) for all h E X iand all i. That is, (2,) is an equicontinuous set of mappings on

xi.

We shall now show that {Z,) converges in [&; B]" to a limit 2 E [&; B ] . for some j . Choose any > 0 and Consider any h E S ; h will be in let M , and x p , c be as in Lemma 4.4-1. By virtue of Lemma 4.3-2, we can select an ho of the form (4) in Section 4.3 such that

llZi(h) - 2dh)II 5 llZi(h - h0)ll

+ II2i(h0) - 2dh0)ll + II2dh0 - h)ll*

For all i and I, the first and third terms on the right-hand side are both bounded by .c1/3 by virtue of Lemma 4.4-1 and (8). Moreover, there exists a constant N > 0 such that, for all i , 1 N , the second term is bounded by ~ ~ because 1 3 of (6). Hence, by the completeness of B, 2 , ( h ) converges to a limit, say 2(h). This defines a mapping 2 from Sj - into B, which is linear since each 2,is linear. Moreover,

=-

11 2(h) 11 B 5

[(h) for all h E X?j-l.But, Lemma 4.3-1 implies that x p , { is a continuous seminorm on S j- Thus, 2 is continuous on every .#',- 1. This being true for every j , we have that 2 E [&; B ] . xp.

4. KERNEL OPERATORS

88

Equation (6) now shows that (1) holds true. Clearly, given 2,331is uniquely determined by (1). On the other hand, Lemma 4.3-2 coupled with (1) of Section 4.3 states in effect that the set 0 of all elements of the form $ v , where 4 E 9 and u E V ,is total in 2.Hence, any member of [&; B] that coincides with 2 on 0 must be identical to 2 on &. This completes the proof of Theorem 4.4-1 except for the proof of Lemma 4.4-1. In order to establish that, we shall need still another lemma.

Lemma 4.4-2. Given any nonnegative integer p valued function ri on R" by

E R",

define the complex-

where qi is thefunction definedin theprecedingproof. L e t j > 1 andlet $ E b e s u c h t h a t $ ( t ) = 1 on K j + l . Then, f o r a l l i > j a n d a l l h E & j - l ,

PROOF.For any therefore

4 E g K ,and

(qi

i

gKj+2

>j , we have that supp q i * 4 c Kj+l and

* 4)(t)= JR" qi(t - T M T )

d~

= /Rn$(t)ri(f - T ) D ~ + " ] ~ (dr. T)

Here, we have used repeated integration by parts. Consider the function T H $(*)ri(*- T ) D ~ + [ ' ] + ( T ) .

It is smooth on R" with values in gKj+2, and its support is contained in K j . By appealing to (9,we may write, for any 4 E g K 1any , u E V j ,and all i>j,

zi(40)= m(qi *

4 9

0)

= lK1%l($(*)ri(. - T),

Dp+'*'4(T)U)dT.

In the last step, we have used Note I1 of Section 1.4 and the separate continuity and bilinearity of +m.

4.5. KERNEL OPERATORS

With i > j still, define the linear operator

89

lion X i by

k i ( h ) 4 J" 'iUl(+(-)ri(. - T), D P + [ l l h ( t ) )ds,

h EX j.

&

(10)

KI

Thus, from (2), we have where s = p + [I], o = [ j , V for some integer v, and N is a constant. This shows that % i is continuous. Obviously, 2, and 2 , coincide on the set { $ u : 4 E g K Iu, E V j }and therefore on its span S. Now, let h E X i - , and ho E S. We may write

II2i(h) - %i(h)IIB 5 IIXi(h - h0)II

+ Il%i(ho - hll.

Upon referring to ( I ) of Section 4.3, to (4) and ( I l), and to Lemma 4.3-2, we see that 2,and %i coincide on X j - Finally, note that the interval K j of integration in (10) may be replaced by K j - when h E X j . This proves (9). 0

-,

PROOF OF LEMMA 4.4-1. We may apply the estimate (2) (with g K j +replaced by g K j fand 2 V iby V j- I ) to (9), where p is chosen equal to [I]. This yields, for all i > j ,

where s = p + [I ] and p = ( j - l ,4. It is not difficult to show that the quantity within the braces is bounded by a constant not depending on i. (This is a result of the fact that diam supp q i is bounded for all i.) Thus, { 2 i } i ,isj an Moreover, for 0 5 i S j , each 2,is equicontinuous set of mappings on Consequently, {Z,},'is ",also equicontinuous on continuous on & ' j - l . X j - I . 0 4.5. KERNEL OPERATORS

Theorem 4.4-1 provides a characterization of the continuous linear operators from g R S ( A )into [ 5 B R , , ; B]"; this is the content of the present section. We start with a special case of Theorem 4.4-1.

Theorem 4.5-1. Corresporzding to every separately continuous bilinear mapping 911 of 9 R t l x g R s ( A ) info B there exists one and only one distribution f~ [ Q R n f S ( A )B; ] such that

9x(4>u,

= ( f ( t %x),

4(r)v(x>>, 4 E g R n

9

QRS(A).

(1)

4. KERNEL OPERATORS

90

PROOF.We choose -Y- = gRs(A), as was done in Example 4.3-1, and then employ (8) of Section 4.3 to set up an isomorphism 3 from &' = 9 R n ( 9 R s ( A ) ) onto g,,+.(A). This induces a bijection from [ Z ;B ] onto [ g R n + = ( A B ) ;] by means of the equation

2(44 = ( f ( h x > , 4 ( t ) v ( x ) > , where 2 E [&'; B ] . (See Figure 4.5-1). But, according to Theorem 4.4-1, every = 2(&v).

9-N can be identified with one and only one such 2 by setting !Ill(&, v) Thus, this theorem is established. 0

Figure 4.5-1

We shall use the right-hand side of ( I ) to define an operatorf., which we call a kernel operator or alternatively a composition operator. GivenfE [ 9 R n + s ( A;) B ] and any v E g R S ( A )we , define the composition productf. v as a mapping on by all 9 E gR,,

(f. 0,4) A

( f ( t , x), 4(t)v(x>>,

t E R",

x E R".

(2)

Thus, f . v maps g R ninto B linearly and continuously. Hence, the kernel operatorf: v ~ f maps . ~g R S ( A )into [ g R m B ];. Clearly, f. is linear. To show its continuity, let @ be a bounded set in g R nThere . exists a compact interval K in R" such that @ c g K .Also, let J be any compact interval in R'. Then, for all v E g J ( A ) ,the mapping{t, x} H 4(t)v(x) is a member of g Kx J ( A ) .Therefore, there exists a constant Q and a nonnegative integer r = { r , , r 2 } E R"" such that

s u ~ l l ( f .v, 4>11 5 SUP Q max su~lI~~C4(t)4x)lll. +E@

O s k s r r,x

4.5.

91

KERNEL OPERATORS

But the right-hand side is bounded by

P max sup((Dk2u(x)(( = f'pr,(u), OSkZ4r2

x

where P does not depend on u. Thus, f.is continuous on Q,(A) for every J . This implies that f.is continuous on g R I ( A ) . We summarize these results as follows.

Theorem 4.5-2. For any given f E [ 9 R n + . ( A ) ; B], the kernel operatorf. is a continuous linear mapping of 9 , , ( A ) into L[a;, B]. Our next objective is to develop a converse to Theorem 4.5-2. We first state a lemnia whose proof is quite straightforward.

Lemma 4.5-1. Let %be a continuous linear mapping of QRs(A) into [Q,"; Define 9'R . from '3 by 9)31(#, u)

'

<%u,

#),

u E QR*(A)?

4 E 9,"

*

B]".

(3)

Then, 9Jl is a uniquely defined, separately continuous bilinear mapping of g R nx QR6(A)into B.

Theorem 4.5-3. If % is a sequentially continuous linear mapping of DR,(A) into [9,"; B]", then there exists a unique f E [gR"t.(A); B] such that 8 = f. on . 9 R s ( A ) . Note. f.is called the kernel representation for '3 and f is called the kernel of %. This theorem implies the precise converse of Theorem 4.5-2 since con-

tinuity implies sequential continuity and the pointwise topology is weaker than the bounded topology. PROOF. As was noted in Section 3.2, the sequential continuity of ! illon Q,.(A) implies its continuity on QRs(A). Therefore, '3 satisfies the hypothesis of Lemma 4.5-1. By virtue of Theorem 4.5-1, the YJl that is uniquely defined ; such that (1) is satisfied. Upon by (3) determines a unique f E [ g R n t S ( A )B] combining (1)-(3), we see that %zu =f.u for all 0 E g R S ( A ) . 0

Example 4.5-1. We determine the kernel representationf. for the operator R",t E R",o7is the shifting operator defined in Section 3.3, andp is a nonnegative integer in R". Note that co, D p is a sequentially continuous linear mapping of QRn(A)into [Q,; B]", so that it must have a kernel representation. co, D p ,where c is an [ A ; B]-valued continuous function on

4.

92

KERNEL OPERATORS

Throughout the following, both t and x are variables in R". Define the distribution g E [ 9 i Z n ( B )B; ] by (9, e)

e(t, t ) dt

E B,

e EQ;~~(B).

R"

Then, define the distribution f by f ( t , x)

c(t)a-,(x)(-

W ' D , p g ( 4 x),

(4)

where a-,(x) denotes a shift of --z in the x direction. Actually, f E [ 9 i Z n ( A ) ; B ] , where q is the (2n)-tuple whose first n components are zero and whose last n components are the components of p . This can be seen from the following equation, where $ E 9 i z n ( A ) :

(f,$> = (dh X I , @CPa,(x)c(t>$(4 Moreover, for any u E BRn(A) and

XI>

4 E BR,, ,

(f.v , 4 > = ( J ( 4 x),4(0u(x>> = JR;(t)4(tp;u(t

- 7 ) dt

= (CC, DPv, 4).

-

Thus, we have shown that carD p =f on g R n ( A ) ,where f is defined by (4). 0 There are a number of extensions of the kernel operators discussed heretofore. Schwartz (1957, pp. 124-1 26) discusses kernel operators that map gRS into gR,,(V), where V is a separated locally convex space. Meidan (1972) into b $ , and, by going to the adjoint considers kernel operators that map gRS C]. An extension operators, he obtains mappings of [b;.; C] into [BRS; onto various spaces of Banach-space-valueddistributions is given by Zemanian (1970~);for example, representations for mappings of [b:.; A ] into [Bin;B ] are presented there. The latter two works are related to still another method of representing certain continuous linear mappings of one space of distributions into another; it was first introduced by Cristescu (1964) and subsequently developed by Cristescu and Marinescu (1 966), Sabac (1965), Wexler (1966), Cioranescu (1967), Pondelicek (1969), Dolezal (1970), and Zemanian (1972a). The basic idea is to assume that we are given a family {y,} of distributions in [ g R nC] ; depending on the parameter x E R" such that the followingcondition is satisfied. For I)&) e (y,, +), 4 H $+is a mapping . define the product u y,, where u E [b,.; C] by of Q R n into b R s Then, 0

4.6.

CAUSALITY AND KERNEL OPERATORS

93

( u 0 y,, 4) (v, II/b). It can be shown that U H U 0 y , is a continuous linear mapping of [b,,; C ] into [ g R nC;] . This implies that it must also be a kernel operator on g R S .

Problem 4.5-2. Prove Lemma 4.5-1. 4.6. CAUSALITY AND KERNEL OPERATORS

Theorems 4.5-2and 4.5-3characterize any continuous linear operator % that maps g R S ( A )into [ g R nB;] . The facts that the domain g R S ( A )of % is a small space with a strong topology [as compared to other spaces, say & ( A ) or &'(A) that one might choose for a realizability theory] and that the range ; with a weak topology implies that of % is contained in a large space [ g R nB] a wide class of operators is encompassed by Theorems 4.5-2and 4.5-3.Any expansion of the domain space or weakening of its topology and similarly any diminution of the range space or strengthening of its topology will in general decrease the class of continuous linear operators under consideration. This is one reason why a distributional approach using testing-function spaces as domains and distribution spaces as range spaces is such a powerful tool in realizability theory. There are many physical phenomena that can be modeled by continuous linear operators on g R S ( A ) However, . many of those phenomena possess still another property, namely, causality. It can be stated loosely by saying that a physical system cannot respond to an excitation until that excitation has been imposed. Or, alternatively, systems cannot predict the future behavior of their excitations. I n this section, we define causality and show how it can be characterized in terms of a condition on the support of the kernel f of any Although our results can be formulated for signals kernel operator % on R" (see Problem 4.6-l),the physically significant situation arises when n = 1. In the latter case, the independent variable t E R = R' is taken to be time and the signals at hand are distributions on the t-axis. We therefore . t , x E R throughout. restrict ourselves to this situation and set 2 = g R ,Also, = f a .

Definition 4.6-2. Let % be an operator mapping a set 3 c [ g ;A J into [g;B ] . '91 is said to be causal on S if, for every T E R,we have that % u , = %u, on the open interval (- co,T ) (in the sense of equality in [g;B ] ) whenever u , , u, E X and u , = u2 on (-a, T). It follows that a linear operator % on a linear space X is causal if and only if %u = 0 on (-a, T ) whenever u E 3 and u = 0 on ( - 0 0 , T ) .

4.

94

KERNEL OPERATORS

Theorem 4.6-1. Let f E [ g R 2 ( A ) B ; ] and let R k { { t , x}: t 2 x}. The kernel operator f is causal on @ A ) ifand only fsupp f c R.

-

PROOF.Assume suppf c Q. Let u E 9 ( A ) be such that u = 0 on (- 00, T). The causality off * will follow once we show that f * v = 0 on (- 00, T ) in the sense of equality in [ 9 ;B ] . Choose an arbitrary 6, E 9 with supp 6, contained in (- 00, T ) . Then,

(f

-

21,

4)

=

x),6,(t)u(x)) = 0

(f(t7

because the support of the function { t , x} H4(t)v(x) does not intersect R and is therefore contained in the null set off. So, truly,f* u = 0 on (- co, T). Conversely, assume that f . is causal on g ( A ) . Consequently, for any v E 9 ( A ) with supp L' c [T, 00) and any 4 E 9 with supp 6, c (- 00, T), we have (f(t,

4, 6,(t)4X))

=

(f

-

0,

6,)

= 0.

(1)

Choose any $ E g R 2 , , ( A )such that supp $ does not intersect R. We shall prove that (.f, $) = 0 and conclude thereby that s u p p f c $2. Since supp $ is a compact set and R is a closed set, the distance between supp $ and R is a positive quantity; that is, inf{ 1 M' - zI : W E supp $, z E R} > 0. Now, we can choose two finite collections { K , ) and ( J , ) of closed intervals i n R2"and a finite collection of members of 9 R 2 , , ( A such ) that the following five conditions hold: K , c J , for each p ; K, 2 supp $; J , does not meet R ; supp $, c K , ; and finally, $ = $, . (See Zemanian, 1965, Section 1.8.) For any fixed j i and all 0 E g J S ( A ) we , may write

u

u

where the constant A4 and the integers q and r do not depend on 0. By Lemma 4.3-2 under the special case of Example4.3-I, given any E E R , ,we can choose h,(h

XI =

c 6,,,d0qt,

v(x)

V

such that the summation is over a finite number of terms, b,,, E 9 , u , ~ , E 9 ( A ) , the support of the function { t , ~ } ~ ~ , , ~ ( t ) o , is , ~contained (x) in J , , and IID,"D,"$,(t,

SUP 1, x E

X)

R"

- h,,(t, x)III

<E/M.

Hence, by ( I ) and (2), II(f7

$,)I1

=

II(f, $, - hJll < E

4.6. CAUSALITY AND KERNEL OPERATORS

95

so that ( f , t+bP) = 0. In view of the fact that the collection { $ P } is finite, we can conclude that

*> 1

(f, =

(f, *J = 0.

P

Thus, supp f c R. 0

Problem 4.6-1. Definition 4.6-1 and Theorem 4.6-1 continue to hold when T ER",and f E [ g R 2 J A ) ;B ] . Show this. we let 9 = gR,,,

Chapter 5

Convolution Operators

5.1. INTRODUCTION

Time invariance is a property possessed by many physical systems. It arises when the structure of the system and the values of its parameters remain fixed with time. As a result, if u ( t ) is the responding signal to some driving signal u(t), then u(t - 7 ) will be the response to u(t - T), whatever be the real number T. In fact, for any operator % of a given Banach system, time invariance is characterized by saying that 8 commutes with the shifting operator 6,. Thus, this property is also called translation invariance. The objectives of the present chapter are to discuss the convolution process in the context of Banach systems and to establish the following basic result: Translation-invariant kernel operators are convolution operators, and conversely.

96

5.2. CONVOLUTION

97

5.2. CONVOLUTION

Here is a theory for the convolution of an [ A ; B]-valued generalized function y on R" with an A-valued generalized function u on R".The resulting convolution product y * u will be a B-valued generalized function on R". Let there be given three p-type testing-function spaces: Y ( A ) , $ = $ ( C ) , and X = X ( C ) .Also, assume that the following three conditions are satisfied.

Conditions E E l . If 4 E X , then, for eachfixed t E R", 4(t E2. With u E [$; A ] and 4 E 2,define $(t)

+ .) is a member of $.

( 4 4 , 4 ( 1+ XI>.

(1)

Then, with vfixed 4 H $ is a continuous linear mapping of .finto Y(A). E3. Under the preceding notation and with 4 E X fixed, v I--, $ is a continuous linear mapping of [$; A]" into Y ( A ) ; moreover, it is uniformly continuous with respect to the G-sets in y . (By this uniform continuity, we mean the following. Given any G-set @ in X and any neighborhood A of zero in #(A), there exists a neighborhood E of zero in [ I A]S ; such that $ E A for all 4 E 0 and all u E E.See Figure 5.2-1 .) Definition 5.2-1. Under Conditions E, the convolution product y * u of any [Y; A ] is defined as a mapping on X by

y E [ Y ( A ) ;B] and any v E

(Y

* u , 4 > 4 640, >,

4 E .f.

(2)

Note that Conditions El and E2 ensure that the right-hand side of (2) has a sense and is a member of B.

Figure 5.2-1

5.

98

CONVOLUTION OPERATORS

Theorem 5.2-1. Given the complex Banach spaces A and B and the p-type testing-jiunction spaces Y ( A ) ,f , and Y ,assume that Conditions E are satisfied. If y E [ Y ( A ); B ] , then the operator y * : u H y * u is a continuous linear mapping of [ f ; A]" into [ Y ;B]".

PROOF.We have already noted that y * u maps X into B. Now, observe that y * u is the composite mapping 4 H II/ H ( y , +). By virtue of Condition E2, we conclude that y * u E [ X ;B ] . As is evident from (2), the mapping U H *~ u is linear. To show that it is continuous, let CD be an arbitrary 6-set in X and consider ?Q(y * v,

II(y

* u, 4 ) l l B

=

II(y,

II/>llB-

OEQ

Given any E E R , , set p {b E B : llbll < E } . Since y is continuous on Y ( A ) , there exists a neighborhood A of zero in $(A) such that y maps A into p. (See Figure 5.2-1.) But, by Condition E3, there exists a neighborhood E of zero in [ f ; A]" such that the composite mapping U H $ H ( U , II/) carries B into p, whatever be the choice of 4 E @. Therefore, yQ(y * v) < E for all u E E. This proves the asserted continuity because any neighborhood Y of zero in [ X ;BIs contains the intersection of a finite collection of sets of the form

BI: -Y*(u> < 4 and we need merely take the intersection of the corresponding Z s to get a neighborhood of zero in [ f ; A]" that is mapped by y * into Y. 0 { u E [.f;

The mapping y * : v H y * u will be called a convolution operator. Any given y E [ $ ( A ) ; B ] generates such an operator so long as Conditions E are satisfied. Problem 5.2-2. Show that Theorem 5.2-1 remains true when the G-topologies of [ f ; A ] and [ X ;B ] are replaced by their pointwise topologies and Condition E3 is replaced by the following statement: U H $ is a continuous A]" into Y ( A ) . linear mapping of [I;

5.3. SPECIAL, CASES

We now list several specific choices of the triplet {$(A), f , X } for which Conditions E and therefore the hypothesis of Theorem 5.2-1 are satisfied.

I.

$(A) = 9 ( A ) , f = 6, X = 9.

As in the preceding section, all testing functions are defined on R",and so it is understood that t E R", 9 = g R nand , d = dRn. Up to now, we have viewed the topology of 9 ( A ) as an inductive-limit topology. But, since this topology is

5.3. SPECIAL CASES

99

locally convex, it must be obtainable from a generating family of seminorms. We shall now determine such a family and use it subsequently. Let E A { E , } $ ~ be a sequence of positive numbers tending monotonically o a sequence of nonnegative integers tending monoto zero and let f A { l v } ~ =be tonically to co.A(1, E ) is defined as the set of all E 9 ( A ) such that, for each v = 0, I , 2, . . . , we have

+

whenever I k 1 I I , and I t I 2 v. By taking the collection of all such A(1, E ) as a basis of neighborhoods for 9 ( A ) , we obtain a locally convex topology 0 for 9 ( A ) . (See Appendix C4.) Note that 0 is generated by the collection {y,, e } of all seminorms on 9 ( A ) defined by

We shall show that 0 is precisely the inductive-limit topology that was previously assigned to 9 ( A ) . Indeed, this is an immediate consequence of the following lemma. We let K be an arbitrary compact subset of R”.

Lemma 5.3-1. A convex set Q c 9 ( A ) is a neighborhood of zero with respect to 0 if and only if it intersects each g Kin a neighborhood of zero in gK. PROOF.Any neighborhood of zero with respect to 0 contains a set of the form A(1, E ) , and the latter in turn clearly intersects every g Kin a neighborhood of zero in g K . Conversely, assume that the convex set R intersects each Q K in a neighborhood of zero in g K .Let K , 3 { t E R“: I tl < v + 2). For each v, there exists an integer I , 2 0 and a positive number qv such that every E 9 ( A ) satisfying supp c K , and ll+(k)(t)ll < q,, for Ikl I I , is a member of R. We can choose the sequence I = {f,} to be monotonically increasing to co. Furthermore, we can choose a sequence {A,};==, c 9 such that &(t) 2 0 and C AV(t) = 1 for all t and supp 1, c { t : v I I tI I v 2). (See Zemanian 1965, Section 1.8.) Then, for every E 9 ( A ) ,

+

+

+

+

where the summation contains only a finite number of nonzero terms. Because of the convexity of n, E n whenever 2’+’AV E SZ for every v. Now, if ll+(k)(t)ll I E , for lkl I I , and I t I 2 v, then, by virtue of Leibniz’s rule for the differentiation of a product,

+

+

100

5. CONVOLUTION OPERATORS

for all t and Ik I I 1, ,where the c, are constants not depending on the choice of 4 E 9 ( A ) . We can choose E A (8,) to be monotonically decreasing to zero and such that c, E , < qv for every v. Thus, the fact that 4 E A(/, E ) implies that 2v+1Av4E R, which, as was noted above, implies that 4 E R. So, truly, R is a neighborhood of zero under the topology 0. 0 We now show that Conditions E are satisfied when 9 ( A ) = 9 ( A ) , f = I, and X = 9. Condition El is obviously fulfilled. To verify Condition E2, we first note that supp $ is a bounded set since supp u and supp 4 are. It is also true that, for every nonnegative integer k E R", = (u(x), p ( t

+ x)).

(2) Indeed, this is true for k = 0 by definition of $. So, assume it is true for any other k, fix t E R", and let At, E R with Atv # 0. Then, $'k'(t)

where

As usual, zI ,denotes an n-tuple all of whose components are zero except possibly for the vth component z, . A straightforward manipulation shows that OAtv + 0 in I as Atv + 0. Since u E [ I ;A ] , ( 3 ) tends to zero. We can conclude by induction that (2) holds true for all k. This implies that $ is a smooth function and is therefore a member of 9 ( A ) . Thus, 4 H $ is a mapping of 9 into 9 ( A ) , which is clearly linear. Furthermore, for any compact set K , supp II/ is contained in another fixed compact set J for all 4 E Q K . Also, since u E [ I ;A ] , ll$(k)(t)ll I M max sup OSkvsp xeIp

I M max sup

I 4(k)(t+ x)l I 4(k)(x)l.

OSk,Sp x6R"

(See Case I1 of Section 3.6.) This implies that 4 H $ is continuous from into 9&4) for every K and corresponding J and therefore from 9 into 9 ( A ) . So, truly, Condition E2 is fulfilled. Finally, Condition E3 asserts that U H $ is linear from [8; A ] into 9 ( A ) and uniformly continuous with respect to the bounded sets in 9.The linearity is clear. To show the uniform continuity, choose an arbitrary seminorm yl, for the topology of 9 ( A ) as defined by (1) and let Q, be a bounded set in and is bounded therein. We have that 9. Thus, is contained in some

5.3.

101

SPECIAL CASES

Now, observe that

+ *):

{~;'$(~)(t

4 ED, IkJ II,,,

It1 2 V ,

v = 0, 1,2, ...}

is a bounded set in 1.Indeed, let Z be any compact set in R". Then, there exists an integer v, such that, for all I tl 2 v, and all 4 E D,

z n supp +(k)(t + .) is a void set. Thus, for only a finite number of v's are there functions 4(k)(t .), where It1 2 v, that are not identically equal to zero on I. Our assertion follows from this fact. Thus, (5) is the same as

+

SUP YI. d$) = SUP Il(& e>ll, e E e

where 0 is a bounded set in 1.This establishes the uniform continuity of 4 + $ and completes the proof of Condition E3. By virtue of Theorem 5.2-1, we have established the following. Theorem 5.3-1. If y E [ 9 ( A ) ;B ] , then mapping of[&; A ] into [9;B ] .

DHY

*D

is a continuous linear

11. $(A) = 9 - ( A ) , 3 = 9 - , 3-= 9 - .

In this part, we take R" = R so that t E R. We first present a direct charac~ a sequence terization of the topology of 9 - ( A ) . Once again, let E { E , , } ~ = be of positive numbers tending monotonically to zero and let I = { I , , } ~ = , be a sequence of nonnegative integers tending monotonically to infinity. Also, let p be an arbitrary positive integer. A(&e , p ) denotes the set of all 4 E 9 - ( A ) such that l14(k)(t)llA Eo - p It , 0 I k _< I, and l14(k)(~)llA v s t , O I k S l v , ~ = 1 , ,.... 2 9

Ev7

The collection of all such A(1, E , p ) is a basis of neighborhoods of zero for a ~ }seminorms generating 3 is defined topology 3 of 9 - ( A ) . A family { Y ~ , & ,of by (6) Y l . &. J4) e max{d4), B(411, where 44) = SUP IIE014(k)(t)IIA -psi OSkslo

and

5 . CONVOLUTION OPERATORS

102

The next lemma shows that 9 is identical to the inductive-limit topology previously assigned to 9 - ( A ) . We let K be an interval of the form (- co, TI, where T E R , and let g Kbe the space of all 4 E 9 - ( A ) with supp 4 c K and supplied with the topology generated by { p i , ,},"=o, where p j , is defined in Case VI of Section 3.6.

Lemma 5.3-2. A convex set R c 9 - ( A ) is a neighborhood of zero with respect to 9 if and only if it intersects each g K in a neighborhood of zero in 9 K

*

The proof is just a modification of the proof of Lemma 5.3-1. We turn to a verification of Conditions E. It is obvious that Condition E l is fulfilled. For Condition E2, we first note that supp u is bounded on the left (Theorem 3.7-4) and that supp 4 is bounded on the right. It follows that $ is an A-valued function whose support is bounded on the right. In fact, for all 4 E g K ,where K = (- co, TI, we have that supp $ c N (- co,T - t ] , where t = inf supp v. That (2) holds and that $ is smooth follows as in Case 1. (Now, however, we have to show that O,, tends to zero in 9-as At +O.) Thus, &I+$ is a linear mapping of g K into g N ( A ) . It is also continuous. Indeed, for any 4 E g K ,for any nonnegative integers k and q, and for each ? € I q , where I4 [-q, co), we have that 4(t + .) € g J , where J = ( - co,T + q]. Since v E [ Q J ;A ] , there exists a constant M and a nonnegative integer p such that sup

~ l $ ( ~ ) ( t ) lI l M

rE I,

max sup

I 4(k+r)(t +x)l.

O s r s p ZEI .€I;

Hence, 4 H $ is a continuous linear mapping of g Kinto g N ( A )and therefore of 9- into 9 - ( A ) . To verify Condition E3, we first note that u b $ is a linear mapping of [9; A ] into 9-(,4). That this mapping is uniformly continuous with respect to all 4 in any bounded set @ in 9-is shown as follows. We know that @ is where K = (- 00, TI as above, and is bounded contained in some space g K , therein. It is also true that SUP 71. E , ,($I

dE@

= SUP 71. E . dE@

,((W,4(f + 4 ) )

where 0 is a bounded set in 9-. This is a consequence of (2) and the fact that the collections

{&;'p)(f + x ) : 4 €@,

- p I t , 0 I k 51,)

5.3

103

SPECIAL CASES

and v s t , O < k < l , , v = l , 2 ,...} are both bounded sets in Q - . (Note that, given any interval of the form I = [q, 03), there is a vo such that I n supp 4(f .) is void for every t 2 vo and all 4 E@.) Equation (7) proves the uniform continuity of U H $ with respect to all 4 E @. Thus, we have established the following theorem. { ~ ; ' 4 ( ~ ) ( t + x ) : 4 ~ @ ,

+

Theorem 5.3-2. If y E [ 9 - ( A ) ; El then U ping o f [ 9 - ; A ] into [9-; El.

H

*

~ u

is a continuous linear map-

111. Y ( A ) = Y ( A ) , f = U ( w , z), where w < 0 < z, X = 9'. We shall examine this case under the restriction that our testing functions are defined on R rather than on R".This, in fact, is all we shall need subsequently. Essentially the same arguments can be used for R",but the notation becomes considerably more complicated. Every I$ E 9' is a member of U ( w , z ) because 4(t) tends to zero faster than any negative power of I t I as I t I + 03, whereas the members of 9 ( w , z ) are allowed to grow exponentially as If +CO by virtue of the condition w c 0 < z. Moreover, it is easy to see that U(w,z ) is closed under the shifting operator. This implies that Condition El is satisfied. Turning to Condition E2, we set up Equation (3) for the A-valued function $, where now Atv is replaced by At and 8, by D. From (4) and the fact that 4 E 9,it follows that, for each nonnegative integerp, Bkq'converges uniformly to zero on R as At -0. In other words, 8,, + O in and therefore in U ( w , z ) as well. But v E [ 9 ( w ,z ) ; A ] . Hence, (3) tends to zero, which by induction establishes (2) and the smoothness of $. To show that $ is of rapid descent, we first observe that 4 E Y implies the existence of positive constants Cq, such that I $ ' P ) ( t ) I 5 Cq,,,/(1 t 2 y , q, p = 0,1, .... (8) Now, let w < c < 0 < d < z, SO that 9' c P c , d . Since the restriction of v to 9c,d is continuous and linear,

I

,

+

11(1

+ r2)'$(j)(t)ll.

I

M max sup I(1 + t 2 ) ' K c , d ( X ) 4 ( k + i ) ( t Osksr xoR

+x)l.

(9)

The quantity within the magnitude signs in the right-hand side is bounded on the { t , x} plane. Indeed, for t 2 0 and x I - t/2, I(1 t 2 ) k C,,(x)4(P)(r , x)l I (1 + t2)'e-d'/2C,, and the right-hand side is bounded for all t 2 0. On the other hand, for t 2 0 and x > - t / 2 ,

+

+

5.

104

CONVOLUTION OPERATORS

and the right-hand side is bounded for r 2 0 when q 2 1. A similar argument for t I0 establishes our assertion concerning the right-hand side of (9). This proves that $ E Y ( A ) . To prove that the linear mapping $I+$of Y into Y ( A ) is continuous, converges in Y to zero. This means that there exist assume that {4y},"=l positive constants C,,,p , which replace C,,, in (8) when 4 is replaced by 4" and which tend to zero as v + 00, q and p being fixed. Then, (9) and the argument following it show that {$,},"= converges in Y ( A ) to zero. Finally, consider Condition E3. Let @ be a bounded set in Y and write SUP SUP Il(1 + t 2 1I IL(1)(t)llA = SUP SUP Il(v(x), (1

4EQtaR

~

E iOc R

+ t2)'4'j)(t + X)>ll".

Condition E3 will be verified once we show that, as 4 traverses @ and t traverses R, (1 -!- f2)14(j)(t+ traverses an 6-set in U ( w , z). To prove the latter, let w < c < 0 < d < z, so that Y c .Yc,d.By Liebniz's rule for the differentiation of a product and by the argument following (9), we have that, for every k, a )

S U P IK,,d(X)D,k[(l xcR

+ t 2 )I 4 ( i ()t + x)ll

is uniformly bounded for all t E R and all 4 €0, which is what we want. The next theorem has hereby been established.

Theorem 5.3-3. If y E [ Y ( A ) ;B ] , then v w y * u is a continuous linear B ] when M J < 0 < z. mapping of [U(w,z ) ; A]" into [Y; Problem 5.3-1. Prove Lemma 5.3-2. Problem 5.3-2. Show that, if y E [ & ( A ) ;B ] , then linear mapping of [Q; A ] into [Q; B ] .

u

UHY

* v is

a continuous

Problem 5.3-3. Show that, if y E [ U ( w , z ; A ) ; B ] , where w < z, then ~ * uy is a continuous linear mapping of [U(M', z ) ; AIs into [U(w,z ) ; B]".

5.4. THE COMMUTATIVITY OF CONVOLUTION OPERATORS WITH SHIFTING AND DIFFERENTIATION

In each of the three cases considered in the preceding section, the convolution operator y * commutes with the shifting operator and differentiation. For instance, in case I, we have the following result.

5.5. REGULARIZATION

Theorem 5.4-1. If y integer in R", then

E

105

[ 9 ( A ) ;B ] , L' E [b;A ] , t E R",and k is a nonnegative

and

in the sense of equality in [9;B ] . PROOF.Since 6, H or6, is an isomorphism on 9, we may write = (At),( ~ ( x )6,(t , (ar(y * u),6,) = (Y * 0, = (Y(t>, (v(x - z), 6,(t + 4)) = (Y

+ x + ?>>) * (arv), 6,).

This establishes (1). A similar argument with c, replaced by Dkestablishes (2). 0

is an important fact. As was indicated in the That y * commutes with introduction, it means that y * is a translation-invariant operator. Moreover, 9 ( A ) can be identified as a subspace of [d;A ] in accordance with Example 3.3-2, and the canonical injection of 9 ( A ) into [ b ;A ] is continuous. Consequently, the restriction of y * to 9 ( A ) is a translation-invariant kernel operator. A major objective of this chapter is to demonstrate the converse; namely, every translation-invariant kernel operator is a convolution operator. For Cases I1 and 111 of the preceding section, we have the next theorem. Its proof is quite similar to that of Theorem 5.4-1. Theorem 5.4-2. Let t E R and let k be a nonnegative integer in R . Then, (1) and ( 2 ) are equalities in [ X ;B ] under either one of the following conditions: (i) y E [ 9 - ( A ) ; B ] , v E [9-; A ] , X = 9- . (ii) y E [ Y ( A ) ;B ] , c E [ Z ( w , z ) ; A ] , where 11'

-= 0 < z, X = 9'.

Problem 5.41. Establish Theorem 5.4-2. Do the same for the convolutions of Problems 5.3-2 and 5.3-3.

5.5. REGULARIZATION

When the convolution operator y * , where y E [ 9 ( A ) ;B ] ,is applied to any * u is a smooth B-valued function. This is called the regularization of y by v, a process we shall now investigate. u E 9 ( A ) , the result y

5. CONVOLUTION OPERATORS

106

Lemma 5.5-1. Let y

E

[ 9 ( A ) ;B ] and u E 9 ( A ) . Set 4 ( Y ( t ) , u(x - 0 ) .

(1)

Then, UH u is a continuous linear mapping of 9 ( A ) into d ( B ) . PROOF.Clearly, u maps R" into B. The argument that was applied to (2) of Section 5.3 can again be used to show that u is smooth [i.e., u E b(B)]and that U'k'(X)

= ( y ( t ) ,u ( k ) ( X - t ) )

(2)

for every nonnegative integer k E R". To show the continuity of the linear mapping D W U , let K and N be arbitrary compact sets in R".Then, { Z ~X

.): x E K , u E 9 , ( A ) } c 9 J ( A ) ,

where J is some other compact set in R". Since the restriction of y to g J ( A )is continuous and linear, S U P l I U ( k ) ( X ) l I B = SUP XEK

II
u(k)(X

X E K

- 2))Il

< sup M max sup I ~ U ( ~ + ' ) ( X

-

XEK

=

- t)11

O s l s r l ~ J

M max sup J l d k + I ) ( r ) \ l . Oslsr f E R "

This shows that P H U is continuous from 9 , ( A ) into b ( B ) for every N and therefore is continuous on 9 ( A ) . 0

Theorem 5.5-1. If y E [ 9 ( A ) ;B ] and u E 9 ( A ) , then, in the sense of equality in [9; B], y * p = ( y ( t ) ,c(- - 0 ) .

Moreorer, r w y

* 1) is a continuous linear mapping of 9 ( A ) into &(B).

PROOF.Let C#J E 9 and consider the manipulations

(3)

5.5.

107

REGULARIZATION

Ail these equalities are obvious except for the one between (4)and (5). To establish that one, let K be a compact interval containing supp 4. Approximate

(6) by the Riemann sum

where 7c

A {Ip)[ll.,5I

is a rectangular partition of K and x p E I , . (We are using the notation of Section 1.4.) Upon applying y to (7), we get Wt),

c4x, -

t)+(x,) vol 1,) =

1 M t ) , 4 x , - t)>4(x,) vol 1,

*

(8)

Because of the continuity of ( y ( t ) ,u( - t ) ) and 4, the right-hand side of (8) tends to (5) as 1 7 I~ + 0. T o show that the left-hand side of (8) tends to (4),we need merely show that the function of t defined by (7) tends in 9 ( A ) to the function of t defined by (6). This can be done in a straightforward manner using the facts that the function

( 4 x) H4 x - t)+(x) has a compact support and that, for every nonnegative integer k E R",

{r,

X} H Dku(x

- r)$(x)

is a uniformly continuous function on R". Thus, (3) holds in the sense of equality in [9;B ] . The second sentence of our theorem has already been established by Lemma 5.5-1. 0

There is another form of regularization that we shall need at one point in Chapter 7. It states in effect that, if y is an [A; B]-valued &-type distribution and v is a smooth A-valued function such that it and all its derivatives decrease exponentially as the magnitude of its independent variable increases indefinitely, then y * u is a member of g ( B ) . Here, B(B) is the space of B-valued smooth functions on R" such that, for each nonnegative integer k E R",

+

Yk(4)

A sup ~ IER"

<

~ ~ ( k ) ( f ) ~ ~ B

We assign to B(B) the topology generated by { y k } , and this makes B(B) a Frtchet space.

5.

108

CONVOLUTION OPERATORS

Theorem 5.5-2. If y E [ g L , ( A ) ;B ] and u E y a , b ( A ) , where b < 0 < a, then, B], in the sense of equality in [Y; Y * v = W),d' - t ) ) , and

VH

y

* u is a continuous linear mapping of Yo, b(B) into B(B).

The proof follows the same scheme as that of Theorem 5.5-1, but the details are rather more complicated. They may be found in the work by Zemanian (1970a, pp. 112-1 14). 5.6. PRIMITIVES

In this and the next three sections, we discuss four concepts concerning A-valued distributions which we shall use in proving that translation-invariant kernel operators are convolution operators. They are the primitives of a distribution, the direct product of an A-valued distribution with a complexvalued distribution, distributions that are independent of certain coordinates, and a change-of-variable formula. These discussions are much the same as those for complex-valued distributions. In our discussion of primitives, we assume onc: again that t E R 4 R' and that 9 = Q R L . Let H denote that subspace of 9 whose elements x have the form x = $(I), where I) E 9. Given any 4o E 9 such that J 40(t)dt = 1, every 4 E 9 has the unique decomposition

4 =4

0

+ XI x is

(1)

where x E H and c = j 4(t)dt. Indeed, clearly a uniquely determined A sLa, ~ ( xdx ) because $ ( t ) = 0 for all suffimember of 9, and so too is I)@) ciently large t . Aprimitive o f f € [9;A ] is any g E [9;A ] such that g(') =f.As we shall see, any f E [9;A] has an infinity of primitives, any two of which differ by a constant member of [9;A]. ( A constant member of [9;A] is a regular member of [9;A] generated by a function of the form t~ a , where a is a fixed member of A. We let a also denote the corresponding constant member of [9;A].) To determine a particular primitivef(-') of a givenfe [9;A], choose 4o as above and assign some value in A to (f(-",+o). We then definef-') on 4 = 4 0 + x by (2) (f(Y 4) 4 c ( f ( - ' ) , 4 0 ) - (f, $). It is not difficult to show that, as { 4 j }--f 0 in 9, the corresponding numerical sequence { c j } tends to zero, whereas { $ j } tends to zero in 9.This implies that f(-')is truly a member of [9;A].

5.6. PRIMITIVES

109

Next, we verify that f(-”is a primitive off; we do this by showing that =f.For any 4 E 9, the decomposition (1) of 4(l)yields c = 0 and = +(l). Therefore,

Cf(-l))(l)

x

= (f, 4>,

which is what we wished to establish. Finally, we demonstrate that any primitive g off differs from the primitive f(-” defined by (2) by a constant member of [9;A ] . For any $ E 9,

So, upon applying g to the decomposition (1) of any
40)

4 E 9, we may write

+ (99 $Y= c(g, 40) - (L$).

Subtracting (2) from this, we get

4,,) and is therefore a constant member of where a = (9,40) - (f(-’), 19;’41. Here is a fact we shall need later on. As usual, nr represents the shifting operator. Lemma 5.6-1. If g

E

[9;A ] and

gars= g for all T E R, then

(994) = a

Jko4

(3)

where a E A is uniquely determined by g. PROOF.We first note that g ( ’ ) = 0 in [9;A ] . Indeed, for every 4 E 9, (g(l),

4 ) = (9,- 4‘”)

= lim ( 9 , (arb- 4)/T> r-0

= lim 0 = 0.

Now, one primitive of zero is zero. Since g is also a primitive of zero, it must differ from zero by a constant member of [9;A ] . In other words, (3) holds. Obviously, there cannot be two different values for a satisfying (3) for all 4. 0 Problem 5.6-1. Prove thatf(-’) as defined by (2) is a member of [9;A ] .

5.

110

CONVOLUTION OPERATORS

5.7. DIRECT PRODUCTS

In this section, t E R" and x E R". Moreover, 4(t, x ) will now denote a function on R"" (and not the value of 4 at { t , x } , which is our usual interpretation). Let f E [ g R n A ;] and g E [ g R S C]. ; We define the direct product f ( t ) x g(x) as a mapping on any 4(t, x ) E g R n by fS

( f ( 0 x g(x), 4(4 4 )A! (fW,( g ( x ) , 4(4 4 ) ) .

(1)

The right-hand side has a meaning since, as a function o f t , ( g ( x ) , 4(t, x ) is a member of gR,, . This can be shown through the same argument as that used in regard to (2)-(4) of Section 5.3. That argument also establishes the following equation : Qk(g(x>, 4(4 X I > = ( g ( x ) , Q W t ,

XI>.

(2)

It follows from (1) that f ( t ) x g(x) is a linear mapping of gRnfS into A . Moreover, a direct estimate of the right-hand side of ( I ) with the use of (2) shows that f ( t ) x g(x) is continuous on .CBJ for every compact set J c R"". Thus, f ( ? > g(x) i 9 R " f S ; A useful fact is the following. The restriction off(t) x g(x) to the set SZ of all testing functions of the form e(t)lC/(x),where 0 E g R n and II/ E g R Suniquely , determinesf(t) x g(x) on all of g R 8 , + This * . is because SZ is total in gfln+*. (See Schwartz, 1966, pp. 108-109.) For this restriction, we have

(f(t>x g(xh W)lC/(X)) = (5 e x g , II/>. 5.8. DISTRIBUTIONS THAT ARE INDEPENDENT OF CERTAIN COORDINATES

Let j

E [9R"+";

t = {tl,

., t n > E R",

* *

r] = {Vl,

{t,I'}

=

A],

* * * 9

?I, E> R",

{t1 , . . ., t n v . . ., ~

We say that j is independent of

9

r]

19

s E >

Rn+"*

if, for every z E R"" with

Z = (0,.

. . , 0, T I , . . . , Zs},

we have that o r j = j [that is, if j ( 5 , r]) is independent of shifts through the r] coordinates]. It will now be shown that such a distribution can be written as

5.9.

A CHANGE-OF-VARIABLE FORMULA

111

the direct product y ( t ) x l(q), where y(t) denotes a member of [9,,; A ] and l(q) denotes the regular distribution corresponding to the function that equals 1 everywhere on R". Let = e(tl, . . . t., ql,. . . , q s - l ) m s ) ,

m,

where 8 E 9 R n + s - 1 and + e E R 1Now, . j is independent of qs. Therefore, $ H ( j , 8 $ ) is a member of [aRl ; A ] ,which is also independent of qs . We may therefore invoke Lemma 5.6-1 to write (j9

ell/> = a(@

I*(V")

4% = a(w1(%),

$(qs))7

where a(8) is a member of A depending on 0 but not on $. Upon fixing $ such that J $(qs) dqs = 1, we see that a(8) = ( q , e), where q E [QR,,+$- I ; A ] is uniquely determined by j . In view of the last paragraph of the preceding section, we can conclude that j is equal to the following direct product: j ( 5 , '1) = q(t1,.

. * , t",q1, . . ., V s - 1 )

x I(%).

Next, we observe that, since j is independent of shifts through qs-l, so, too, is q. Indeed, let ur be a shift through the coordinate qs-l only and let 8 and $ be as above. Then, ( 4 , e>(l, $) = ( j , ell/> = ( U r j ,

e$> = (q, a-re>(l, $)*

Since this holds for all such 8 and I), q = u,q. Therefore, by applying the argument of the preceding paragraph to q, we see that there exists a unique p E [ 9 R n + = - 2 ; A ] such that i(t,?)=P(t1,...,t,,11,...,rls-2) x 1(qs-A x l(qJ

We may also write l ( ~ " - ~x ) l(qJ = l ( ~ " - ~q,). , Continuing in this way, we arrive at the following result.

Theorem 5.8-1. Let j E [ E R n +A=];be independent of q . Then, there exists a unique y E [ 9 R n ; A ] such that

At, 4 = Y ( t ) x

w

5.9. A CHANGE-OF-VARIABLE FORMULA

In this section, 9 = QR,, . Let z, 5 E R" and set z = U c , where U is a nonsingular linear transformation on R". Thus, U can be represented by a nonsingular n x n matrix of real numbers and has an inverse U - '. Moreover, IU I = I U 1, where I U I denotes the magnitude of the determinant of U.

-'

-'

5. CONVOLUTION OPERATORS

112

Given any f E [9; A ] , we let f ( U [ ) denote the linear mapping from 9 into A defined by ( f ( 4 ,I u I -Wu -'z>>, 0 E 9. (1) f(UC) is continuous. Indeed, if the sequence (0,) tends to zero in D, then, clearly, { 1 U I -'B,(U -'z)} tends to zero uniformly for all z E R" and the supports of these functions remain contained within a fixed compact set. Moreover, the chain rule for differentiation (Kaplan, 1952, p. 86) shows that the same is true for each fixed-order differentiation of these functions. Thus, I Ul -'0,(U-'z) + O in 9 as v 3 oc). The continuity of f ( U [ ) now follows A]. from (1). We conclude that f(UC) E [9; An alternative form for (1) can be obtained by setting $(z) = O(V-'z):


w>= ,I u I$(UC)>.

(2)

5.10. CONVOLUTION OPERATORS

We are at last ready to establish the condition (namely, translation invariance) under which a kernel operator from 9 ( A ) into [9; B ] becomes a convolution operator. Once again, it is understood that 9 = BR7and 9 ( A ) = 9 R 4 4

Dejinition 5.10-1. Let S and I be spaces of functions or distributions on R" and assume that S and I are closed under the shifting operator 0,. A mapping % from 5F into I is called translation invariant if u,%f = %u, f for every f E S and every z E R". In certain subsequent discussions, we shall call the members of a given family of operators translation varying to indicate that the condition of translation invariance is not imposed even though certain members of that family may be translation invariant. Thus, translation-invariant operators are considered to be a special case of translation-varying operators. When n = 1 and the space R' on which the members of S and I are given is interpreted as the time axis, the adjective translation-invariant is commonly replaced by time-invariant and translation-varying by time-varying. Assume now that % is a continuous linear mapping of 9 ( A ) into [9; B]". By Theorem 4.5-3, % is a kernel operator; that is, 'ill=f on 9 ( A ) , where f E [9RIm(A); B ] . If, in addition, % is translation invariant, we may write for every 4 E gYv E 9 ( A ) , and z E R",

< f ( 4 XI, 4(t + Mx))= <mu, a-74) = <%92U, = <%%V,

4) =( f 0 Y

X),

4) 4(t)v(x - TI>.

5.10. CONVOLUTION OPERATORS

113

[A; B)). So, we may replace v

By Theorem 3.5-I,Jis also a member of [~R2n; by any w E ~ to obtain


I(t, x) =/(t - or, x - or) for every or E R". We now make the change of variables t = this can be written as z = UC, where

z=

[~],

C=

~

(1)

+ '1, x = '1. In matrix notation,

[~].

Here, 111 denotes the n x n matrix, U is nonsingular, and I U I = 1. Thus, by Equation (2) of Section 5.9,


'1), t/I(~

=== (j(~,

+ '1, '1»,t/I E ~R2n

(2)

where j(~, '1) ~/(~ + '1, '1) represents a member of [~R2n; [A; B]]. We now shift j in the '1 direction through the increment or E R n and appeal to (I) and (2): <j(~,

'1 - or), t/I(~

+ '1, '1» ===
This shows thatj(~, (J traverses ~R2n According to [~; [A; B]] or, definition of the

or, x - or), t/I(t, x»
+ '1, '1».

'1) is independent of '1 because, for (J(~, '1) ~ t/I(~ + '1, '1), as t/I traverses ~R2n. Theorem 5.8-1, j(~, '1) = y(~) x 1('1) for some unique Y E equivalently, y E [~(A); B). Thus, by (2) again and the direct product,



= \y(~), Set t/I(t, x) = r/J(t)w(x), where written as

<91(wa),

r/J, w E ~,

x 1('1),

tnt/l(~

t/I(~

+ '1, '1»

+ '1, '1) d'1)'

and let a E A. Then, (3) can be re-

r/J) = /J: (wa), r/J) =
\y(~),

tnw('1)ar/J(~

(3)

+ '1) d'1) =
114

5. CONVOLUTION OPERATORS

But Lemma 3.5-1 asserts that the set of elements of the form wa is total in 9 ( A ) . Moreover, both % and y * are continuous and linear on 9 ( A ) . Thus, under our assumptions on %, we can conclude that % = y * on 9 ( A ) , where y is unique. Conversely, Theorems 5.4-1 and 5.5-1 show that every convolution operator y * possesses the properties assigned to %. We summarize all of this as follows. Theorem 5.10-1. % is a continuous, linear translation-invariant mapping of 9 ( A ) into [9;B]" ifand only ifthere exists a y E [ 9 ( A ) ;B ] such that % = y * on 9 ( A ) . y is uniquely determined by 92,and conversely. An alternative proof of this that does not make use of the kernel theorem is given by Zemanian (1970a, pp. 118-119). Also, we again point out that a linear mapping % on 9 ( A ) is continuous if and only if it is sequentially continuous. Corollary 5.10-la. A continuous linear translation-invariant mapping %from 9 ( A ) into [9;B]" can be extended by means of its convolution representation % = y * onto the space [ f ; A ] and the range of the extended mapping will be contained in [ X ;B ] so long as y E [$(A); B ] and the spaces Y ( A ) , f , and X are p-type testing-function spaces satisfying the Conditions E stated in Section 5.2. Furthermore, if 9 ( A ) is dense in [ f ; A ] , this extension is unique in the sense that no other continuous linear mapping of [ f ; A ] into [ X ; B ] can coincide with % = y * on 9 ( A ) . It is a fact that 9 ( A ) is dense in [&; A] (see Problem 5.10-1). Consequently, for every y E [ g ( A ) ;B ] , y * has a unique extension onto [&; A]. Moreover, with 6 E [&; C ] denoting the delta functional (6 is also called the unit impulse), we can write y * 6 = y in the sense of equality in [a;[A; B ] ] .Thus, we may interpret y as the response of y * to the input 6. Because of this, we shall refer to y as the unit-impulse response of % = y * . The Laplace transform Y of y (which is defined in the next chapter) is called the system function for %. If it happens that supp y is bounded on the left or, equivalently, if y E [ 9 - ( A ) ; B] (see Theorem 3.7-4), y * can be extended onto [9; A ] . Moreover, 9 ( A ) is dense in [9; A], and therefore this extension of y * is unique. Similarly, if supp y is bounded or, equivalently, if y E [&(A);B ] (see Theorem 3.7-3), y * has a unique extension onto [9;A ] according to Problem 5.3-2 and the density of 9 ( A ) in [9;A ] . Problem 5.10-1. Prove that 9 ( A ) is dense in [&; A]. Hint: Choose a sequence { O j } c 9 such that Oj dt = 1, Oj 2 0, and supp O j = [ - j - ' , j - ' ] .

5.1 1. CAUSALITY AND CONVOLUTION OPERATORS

115

Show that 8, --t 6 in [&; C ] ,f * 8, E 9 ( A ) for f E [&; A ] , and there exists a compact set K such that suppf* 8, c K for every j. Conclude from this that f * 8, +fin [&; A]. Problem 5.10-2. Prove that 9 ( A ) is dense in both [9; A] and [9; A].

5.11. CAUSALITY AND CONVOLUTION OPERATORS

As was mentioned in Section 4.6, the concept of causality (see Definition 4.6-1) has physical significance when the signals at hand are defined on the real line. For this reason, we again restrict t and x to R Li R' and set 9 = gR1. Our objective now is to establish a condition on supp y that characterizes the causality of the convolution operator y *

.

Theorem 5.11-1. Let y E [Q(A); B ] . The convolution operator y 9 ( A ) if and only ifsupp y c [ 0, a).

* is causal on

PROOF.Assume y * is causal in Q(A). The regularization formula (Theorem 5.5-1) is

(v * u)(t) = O ( X ) , u(t - XI>,

?JE

%A).

(1)

If supp v c (0, a),then, by causality, this regularization is equal to zero for t < 0. But, given any 4 E 9 ( A ) with supp 4 c (-a, 0), we can choose D and t such that 4(x) = u(t - x). Hence, (y, 4 ) = 0 for every such 4, which means that supp y c [0, co). Conversely, assume that suppy c [O, co). If supp u c [T, co), then supp u(t - .) c (- co, t So, supp u(t - -) does not meet supp y when t < T. Hence, y * u = 0 on (- co, T), and by the linearity of y * , this implies the causality of y * on 9 ( A ) . 0

a.

Theorem 5.11-2. Assume that #(A), f , and X are normal p-type testingfunction spaces satisfying the Conditions E given in Section 5.2. Let Y E [#(A); B ] . If y * is causal on 9 ( A ) , then y * is causal on [ f ; A ] . PROOF.Since #(A), f , and X are normal, it follows that y E [ 9 ( A ) ;B ] , A], and y * u E [ X ;B] are distributions. According to the preceding theorem, supp y c [0, co). Assume that supp u c [T, 00). We wish to show that supp y * u c [T, 00) also. Let 4 E 3f with supp 4 c (- 00, T). Then, D E[f;

<w,4(* + 4 ) E 4 4 ) .

116

5. CONVOLUTION OPERATORS

Moreover, SUPP(V(X), 4(- + x>>= (-a, 0) because, for any fixed t > 0, $(x) H4(t + x) is a shift to the left, so that the supports of u and +(t + .) do not meet. Consequently,

0,* 094) = ( Y ( 0 , ( W ,40 + 4)) = 0. Since this is true for all 4 as chosen above, supp y * v c [ T, 00).

0

Problem 5.11-1. Do the results of this section extend to convolution operators whose signals are given on R"? Problem 5.11-2. Construct another proof of Theorem 5.1 1-1 by deriving it as a special case of Theorem 4.6-1.

Chapter 6

The Laplace Transformation

6.1. INTRODUCTION

Much of realizability theory for a translation-invariant operator consists of criteria imposed upon its system function, which by definition is the Laplace transform of the unit-impulse response of the operator. Consequently, the Laplace transformation on operator-valued distributions is essential to our purposes and is our next topic. The scalar version of the results of this chapter is given by Zemanian (1968a, Chapter 3). Throughout this chapter, we restrict our attention to testing functions and distributions on the real line R. Thus, 1 E R and 9 = gR1. 6.2. THE DEFINITION OF THE LAPLACE TRANSFORMATION

A natural testing-function space to use in discussing the Laplace transformation of [A; B]-valued distributions is U ( w , z ; A) (see parts IV and V of 117

118

6. THE LAPLACE

TRANSFORMATION

Section 3.6 with m set equal to [ a ] ) .We repeat its definition here for the case where its members are defined on R rather than on R". Given any c, d E R, set

9 , , , ( A ) is the linear space of all A-valued smooth functions on R such that

sc, d(A) is a FrCchet space under the topology generated by the multinorm { Y E , d , k%= 0 * Now, let {cj}T=, be a strictly decreasing sequence in R tending to w, where either w E R or w = - a.Similarly, let {dj}?= be a strictly increasing sequence in R tending to z, where either z E R or z = co. By definition, 9 ( w , z ; A) is the inductive limit of the YC,,d,(A).It is not a strict inductive limit but is, on the other hand, a normal p-type testing-function space. As usual, we set 9 ( w , z; C) e 9 ( w , z). For each fixed [ E C and nonnegative integer p, t H tPe-Sris a member of 9 ( w , z) if and only if w < Re [ < z. As was indicated in Section 3.7, [ Y ( w ,z ; A ) ; B ] is a subspace of [ 9 ( A ) ;B ] and is therefore a space of [A ;B]-valued distributions. According to Theorem 3.7-2, [ 9 ( w , z ; A); B] can be identified with [ 9 ( w , z); [A; B ] ] by means of the bijection f w g defined by (9, +>a = < f , *a>,

+

(1)

where f E [ 9 ( w , z ; A ) ; B ] , g E [ 9 ( w , z ) ; [ A ;B ] ] , E 9 ( w , z), and a E A. This identification is henceforth understood, and we will use the same symbol to denote both f and g . A y E [ 9 ( A ) ;B ] is said to be Luplace-transformable if there exist two elements q1 and q z in the extended real line [- co, co] such that q1 < q z , y E [ 9 ( q 1 , q 2 ;A ) ; B ] , and y 4 [ 9 ( w , z ; A ) ; B ] if either w < q1 or z > qz . The open strip

Q,,A{CEC;

q1 < R e C < q Z )

will be called the strip of dejinition for the Laplace transform of y . By the aforementioned identification, y E [ 9 ( q l , qz); [ A ; B ] ] also. Thus, we may define the Luplace transform Y of y as a mapping of R, into [A; B ] by

Y(C)4 W),e - 9 , c E sl,. (2) The Laplace transformation f? is by definition the mapping y w Y. Whenever, we write "y E [ 9 ( A ) ;B ] and (f?y)([) = Y([)for [ E a,,," it is understood that y is a Laplace-transformablemember of [ 9 ( A );B ] and R, is the strip of definition for f?y A Y.

6.3.

ANALYTICITY AND THE EXCHANGE FORMULA

119

We may replace A by C and identify [C; B] with B, in which case (2) becomes the definition of the Laplace transform Y of a B-valued distribution and Y becomes a B-valued function on R, . One further comment should be made here. Our definition of a Laplacetransformable distribution can be simplified somewhat. We could simply say that g E [ 9 ( A ) ;B] is Laplace-transformable iff E [9(al,0,; A); B] for some ol,o2 E [- 00, 001 such that ul < 02.This is because there will then exist a unique pair ql, q2 E [- 00, co] with q1 I; o1 and o2 I; qz and a unique extension f of g that satisfies our previous definition for a Laplace-transformable distribution (Zemanian, 1968a, pp. 55-56). It will tacitly be assumed throughout this book that every such g has been replaced by its extensionf, and we will on occasion use the simpler definition of a Laplace-transformable distribution. Problem 6.2-1. Show that, if u I; w and z I u, then 9 ( w , z) is a dense subspace of 9 ( u , u) and the canonical injection of 9 ( w , z) into 9 ( u , u) is continuous. As a result, [ 9 ( u , u ; A); B] is a subspace of [ 9 ( w ,z ; A); B]. Problem 6.2-2. Show that the shifting operator is an automorphism on [ 9 ( w , z; A); B] under either the pointwise topology or the 6-topology. 6.3. ANALYTICITY AND THE EXCHANGE FORMULA

Theorem 6.3-1. If y E [ 9 ( A ) ;B] and (2y)(c)= Y(c) for c E R, , then Y 4 2 y is an [ A ;B]-valuedanalytic function on Ry,and, for each nonnegative integer k, Y ( ~ ) (=[ )(y(t), (- t)ke-c'), c E a,. (1) This is proven by induction on k in the usual way. Upon fixing c E R,, we write

and then show that $As tends to zero in 9 ( q l , q2) as AC +O. The Laplace transformation converts convolution into multiplication. This is indicated by Equation (2) in the next theorem, which is called the exchange formula. Theorem 6.3-2. Z f y E [ 9 ( A ) ;B]and(2y)(c)= Y(c)fore E R,, $0 6 [ 9 ; A ] and (2u)(c) = V([)for [ E R, ,and if Ryn R, is not empty, then y * u exists in

120

6. THE LAPLACE TRANSFORMATION

accordance with Problem 5.3-3,where now the open interval (w, z ) is the intersection of R, n R, with the real axis. Moreover, for each ( E R,, n n,, (2.Y * v ) ( O = Y(CI(V(0.

(2)

Note. The right-hand side is a B-valued analytic function of ( E R, n R, because Y is [A ; B]-valued and V is A-valued.

PROOF.According to Problem 6.2-1, y e [ 9 ( w , z ; A ) ;B] and U E [ 9 ( w ,2 ) ; A ] . Hence, y * v exists in the sense of Problem 5.3-3.Moreover, for each ( E Q, n R,, we may write (2.Y * v)(O = (YO), (W,e - s ( ' + x 9 ) = cv(t>,e - C ' m > .

Since V(r>E A, we may invoke (1) of Section 6.2 to equate the last expression to V(o= Y(C)V(C). 0 Problem 6.3-I. Prove Theorem 6.3-1.

Problem 6.3-2. Again let y E [Q(A);B] and (2y)(() = Y(() for ( E R,. Establish the following two operation-transform formulas, where k is a nonnegative integer and a, is the shifting operator :

(2Y'k')(o = Y(0,

r

'(0,

(

rk

(2aTY)(C)

= e-CT

E

a,

(3) *

(4)

6.4. INVERSION AND UNIQUENESS

Two [A ;B]-valued distributions having the same Laplace transforms with identical strips of definition, say {( E C: q1 c Re ( < q2}, must be equal as members of [9(ql, q 2 ; A); B]. This is the uniqueness property of the Laplace transformation. It is an almost immediate consequence of the following inversion formula. Theorem 6.41 (Inversion). If y E [Q(A);B] and (2y)(() = Y(() for E C: q, < Re c q2}, then, in the sense of convergence in [ W A );BY,

(E2 !, = {(

y(t) = lim (1/27r) I+ 00

J' Y(c)es' do, -r

where o = Im ( and Re ( isfixed such that q1 c Re [ < q 2 .

6.5. A CAUSALITY CRITERION

121

The proof of this theorem is the same as that of Theorem 3.5-1 given by Zemanian (1968a). Even though we are now dealing with A-valued testing functions and [A ;B]-valued distributions, no alteration in the proof is needed except for the replacement of certain magnitude signs by norm symbols. Theorem 6.4-2 (Uniqueness). Let y , and y2 be members of [ 9 ( A ) ;B] and let (i!y,)(c) = Y,(c) for s E R,, and j = 1, 2. Also, assume that Ryl n R,, is not void and that Yl = Y2 on R,, n a,. Then, y , = y2 in the sense of equality in [ 9 ( w ,z; A ) ; B], where (w, z) = QYl n R,, n R.

PROOF. By Theorem 6.4-1, y , coincides with y2 on 9 ( A ) . But 9 ( A ) is dense in Y ( w , z ; A), and the restrictions of yI and y 2 to 9 ( w , z ; A) are continuous. Hence, y , and y2 coincide on U(w,z; A) as well. 0 6.5. A CAUSALITY CRITERION

If the unit-impulse response y of a convolution operator happens to be Laplace-transformable, then the causality of y * can be' characterized by a growth condition on the Laplace transform Y of y . Theorem 6.5-1. Necessary and sufficient conditions for Y to be the Laplace transform of a Luplace-transformable distribution y E [ 9 ( A ) ;B] with supp y c [0, co) are that there be a half-plane C, {c E C: Re c > 0 ) on which Y is an [A;BJ-valuedanalytic function and there be a polynomial P for which I I y ( c ) 1 1 [ A ; E ] sp((c1),

Rec'0.

(1)

When this is the case, y E [U(a, 00 ;A ) ;B] and the strip of definition of 2 y contains the half-plane C, .

PROOF.Necessity. Since supp y c [0, 00) and y is Laplace-transformable, it follows that y E [U,, d(A);B] for some 0 and every d. We are free to choose 0 > 0 because, for 0 < 0, 9,,d(A) 2 L?', d(A). Thus, Y A 2 y exists on some half-plane bounded on the left and containing C,. Now,let ;Z be a smooth, real-valued function such that ;Z(t) = 0 for t < - 1 and A(t) = 1 for t > - 3. Arbitrarily choose a c E C such that Re c> 0 > 0. Then, for every d > Re c, we have that t H e-[' is a member of .Ya,s(A) and I&, d ( t ) 5 ear for all t. Therefore, there exists a constant M > 0 and a nonnegative integer r such that

I1 y(c)II = II(Y(t>,

e-">II

IM sup

sup

OSkSr t t

- 1/ICI

I e-"'Dk[A( 1 C 1 t)edC']I.

(2)

6.

122

THE LAPLACE TRANSFORMATION

Note that the right-hand side is independent of d and that (2) holds whenever Re [ > u. Moreover, remains bounded for all t and [ such that Re [ > cr > 0 and t 2 - 1/ I [ 1. As a result, the right-hand side of (2) is bounded by a polynomial in 1 [ I. Sufficiency. We first establish a lemma concerning classical Laplace transforms.

Lemma 6.5-1. Assume that, on the halfplane C,, A {[ E C: Re [ > cr}, G is an [ A ; B]-valued analytic function and

IIG(C)II

MI51- 2 ,

(3)

where M is a constant. Sef

I

00

g(r)

(1/2n)

-m

G(c + iw)e(c+iw)t do,

c

> u.

(4)

Then, g is a continuous [ A ;B]-valuedfunction for all t and is independent of the choice of c > u. Moreover, g(t) = 0 for t < 0 , and m

.

g(t)e-5' dt,

G([)=

[ E C,.

0

Infact, G is the Laplace transform of the regular distribution f generated by g.

E [9(u,

co ;A ) ;B ]

PROOF.That g does not depend on the choice of c > u follows from (3) and Cauchy's theorem (Theorem 1.8-1). Its continuity follows from the continuity of tHe-ctg(t), which in turn can be shown by writing

1 I 2n -

"

1

-m

IIG(c

+ iw)ll

and noting that the right-hand side tends to zero as x -P 0. Similarly, it can be seen that e-"g(t) is bounded on the domain { { t ,c}: t E R, c > o}. This implies that g(t) = 0 for t < 0. Indeed, if g(t) # 0 at some f < 0, then e - c t ~ ~ g (can t ) ~be ~ made arbitrarily large by choosing c large enough. We now observe that o ~ G ( + c iw) is a smooth [ A ;B]-valued function. Hence, by virtue of (3), the standard proof for the Fourier-inversion formula (see, for example, Zemanian, 1965, Section 7.2) can be applied to obtain

6.5. A CAUSALITY CRITERION

123

(5) from (4).Finally, g generates a regular distribution f E [Y(a, co ;A); B] by means of the definition

jm s(tW(t>dt,

(f,4 )

-m

4 E %T

00 ;

4

because, for any a, b E R with a < a < b < 00, we can choose c with a < c < a and then write, for all 4 E Y,,,b(A),

IIU, 4)llB

)I

= 1000e-'fg(t)e'c-a)feaf4(t) d t

/I

1 e(c-a)'dt. 00

Isup Ile-cfg(t)ll sup Il&,,b(t)4(t)ll t

f

0

Thus, f E [Y,,, ,,(A) ; B ] , and therefore f E [&'(a, co ;A); B]. Hence, (5) is equivalent to G ( [ ) = (i?f)([) for [ E C, . Our lemma is established. The sufficiency part of Theorem 6.5-1now follows readily. By the condition (l), there exists a positive integer m such that G([) 4 Y([)/["'satisfies the hypothesis of the lemma. Therefore, G = !i?Jwhere f E [L?(o, co ;A ) ; B ] . According to Problem 6.3-2, Y([)= ("G([) = (i?f ("))([) for [ E C, . We conclude by noting that c suppf = supp g c [O,

a).0 Since the condition supp y c [0, co) is equivalent to the causality of y * (Theorem 5.1 1-1), the last theorem provides the following causality criterion promised in the title of this section. suppf

Corollary 6.5-la. Let y E [ 9 ( A ) ;B ] and Y([)= (i?y)([)for [ E ay.The convolution operator y * is causal on 9 ( A ) if and only i f there exists a havplane {( E C : Re [ > a} on which Y is analytic and 11 Y([)II is bounded by a polynomial in I [ I. In this case, y is causal on [9; A ] as well. For the last sentence, see Theorem 5.11-2.

Chapter 7

The Scattering Formulism

7.1. INTRODUCTION

So far, we have seen that the linearity and continuity of an operator W from 9(H) into [Q; HI is equivalent to a kernel representation for W (Theorems 4.5-2and 4.5-3),that W is in addition translation invariant if and only if it has a convolution representation (Theorem 5.10-1),and that the causality of W is characterized by a support condition on the kernel or unit-impulse response of these representations (Theorems 4.6-1 and 5.11-1). However, we have not as yet developed any results arising from energy considerations. This is our last objective. A Hilbert space is the natural framework in which to examine questions concerning energy and power flow, and consequently we will be concerned henceforth with Hilbert ports. The net energy e(Z) absorbed by a Hilbert port over some time interval I c R is described in two distinct ways, de124

7.2. L

P -DISTRIBUTIONS ~ ~ ~ ~

125

pending on whether the scattering formulism or the admittance formulism is chosen. In the former case,

where 2B is the scattering operator for the Hilbert port. In the latter case, e ( l ) = Re

J’I (%u(t), u(t)) dt,

(2)

where % is the admittance operator. If the Hilbert port has no energy sources within it, it cannot impart more energy to its surroundings during the time interval IT { t : - co < t T}, where T I co, than it has received, and thus, e(lT)2 0. This property is called passivity. However, one should (and we will) distinguish between the passivity conditions arising from (1) and (2) because they affect the representations for 2B and % in very different ways. Also, the cases where T = 03 and Tis finite but arbitrary will also be treated separately, the former one being a weaker assumption than the latter. We will take up the scattering formulism in this chapter and the admittance formulism in the next. Passivity is a strong assumption. For example, linearity and passivity imply causality and continuity. This is fairly obvious in the scattering formulism (see Section 7.3) but is by no means obvious in the admittance formulism (see Sections 8.2 and 8.3). Similarly, linearity, continuity, time invariance, and causality do not ensure that the unit-impulse response g of the operator at hand is Laplace-transformable. However, g is indeed Laplace-transformable when the assumption of passivity is added. Much of the subsequent discussion will be directed toward the Laplace transform of g and will result in the so-called frequency-domain formulation for our realizability theory. Throughout our discussion of Hilbert ports, we adopt the natural assumption that the signals at hand are functions or distributions on the real time axis. Thus, in this and the next chapter, t E R and 9 = Q R 1 .

-=

7.2. PRELIMINARY CONSIDERATIONS CONCERNING L,-TYPE DISTRIBUTIONS

We start by establishing certain properties of the distributions in [ 9 , , ( A ) ; B] and [ 9 , * ( A ) ; B]. These two spaces were discussed in Section 3.8. Lemma 7.2-1. I f f € [ g L 1 ( A )B] ; and if s u p p f c [0, co), t h e n f e [ Y ( O , co; A ) ; BI.

7. THE SCATTERING FORMULISM

126

Note. According to Lemma 3.8-3 [ g L 2 ( A )B; ] c [ g L , ( A ) ;B ] , and hence this lemma also holds for all f E [9&l); BI.

PROOF.Let c, d E R with 0 c c < d c co.Let 4 E Zc, d ( A ) . Finally, let 1 be a smooth, real-valued function on R such that A(t) = 0 for - 00 c t < - 1 and A(t) = 1 for c t < 03. Then, A+ egL1(A). Indeed

-+

and the right-hand side is finite because Il&')(t)(j I Ne-" on - 1 c t < co for some constant N . As was indicated in Section 3.3, (f, 4) depends only on the values that 4 assumes on some arbitrarily small neighborhood of suppf. Therefore, we can extend the definition off onto Z c , d(A) by means of the equation (f, 4)

n4>,

= (f,

4

sc,d(A)*

Sincef E [ g , , ( A ) ; B ] ,there exists a constant M > 0 and a nonnegative integer r such that

In view of (I), the right-hand side is bounded by

Hence,f E [.Yc,d ( A ) ; B ] .Since this is so for every c, dsuch that 0 < c c d < co, we have f E [Z(O,co ;A ) ; B ] . 0

Lemma 7.2-2. Let f E [ 9 ( A ) ;B ] andp E R with 1 < p c co. Zf

:/

Il(f*

$)(t)llBP

dt

W

for every $ E 9 ( A ) , then f E [QL,(A); B ] , where q = p / ( p - 1). PROOF.Set

4

Let $ E 9 ( A ) and 4 E 8. In the following, $ ( t ) 4 $(-t). Then, f * E &([A;B ] ) and f * $ E b ( B ) according to Theorems 3.5-1 and 5.5-1. Moreover, <4<x),$0 + XI> = <$(X)Y

40 + XI>,

7.2. L p - DISTRIBUTIONS ~ ~ ~ ~

127

This inequality is Holder's (Appendix G20). Because of (2), the left-hand side traverses a bounded set of real numbers when 4 traverses B and t,b andfare held fixed. Thus,f* 6 traverses a bounded set in [ 9 ( A ) ;B]'. Now, let K and N be any two compact intervals in R such that K cfi.By Lemma 3.7-1, there exists a constant M > 0 and an integer m 2 0 such that, for all I E 9&4), supll(f* 6, I>ll* < max suPllA(k'(t)llA. (3) 918

Osksm toR

Also, by Lemma 3.2-2, for any 8 E giR"'(A),we can find a sequence {I,}T= c 9&4) such that I j -,8 in g N m ( A ) This . fact and (3) imply that f * 6; has a unique extension as a member of the space [gK"'(A);B ] . (See Lemma 3.4-5.) This extension, which we also denote by f * 6, satisfies (3) with I now allowed to be any member of QKm(A).Therefore, as q5 traverses p, f * 4 traverses a bounded set in [ g K m ( A )B]". ; Moreover, for any 8 E gK"'(A),we may write

(f* 8, 4 ) = , where the usual definition of distributional convolution is used. Thus, f * 8, 4 ) traverses a bounded set in B as #I traverses B. Next, set 8 = xu, where x E gK"'and a E A. We obtain

<

(f* 8 9 4 ) = (f* x, +>a, where in the right-hand side, f E [9;[ A ; B ] ] and (f* x, 4 ) E [ A ; B ] . Since the left-hand side traverses a bounded set in B as #I traverses B, we have, from the principle of uniform boundedness (Appendix D12), that (f* x, q5) traverses a bounded set in [A ; B ] . Hencef * x is a continuous linear mapping of 9, supplied with the topology induced by g L qinto , [A; B ] . [See Appendices C8 and D2(iii).] But 9 is dense in g L qand , thereforef * x has a unique extension as a continuous linear mapping of gLQ into [A ;B ] . Upon denoting this extension by f* x, we obtain f * x E [aL,; [ A ; B ] ] for every x E 9,"'. Finally, let y E 9 be such that y = 1 on a neighborhood of 0 in R.Let K be a compact interval in R such that supp y c K. Set n = m + 2, where m is the integer corresponding to N =I k as above. Set

7. THE SCATTERING FORMULISM

128

+ c, where 5 E g K .Thus, f =f * 6 =f * D"(yJ,) + f * c = D " ( f * YJ,)

Then, S = D"(yJ,)

+f* 5.

According to the preceding paragraph, bothf* yJ, and f * { are members of [gLq; [ A ; B ] ] . But then, so too is S, because [gL,;[ A ; B ] ] is closed under D". By Theorem 3 . 8 - 1 , f ~ [QLa(A);B]. 0 It is worth mentioning here that the converse to Lemma 7.2-2 is not true, as can be seen through the following counterexample due to L. Schwartz. Let p = 2, A = C,and B = L , . Also, let f be the identity operator on L , . Then, clearly, the restriction o f f t o gL2is a member of [gL2; L , ] . But, for any E 9,

*

(f**)(t>

= ( f ( x ) , * ( t - 4) = * ( t

- .). The right-hand side is a mapping of the real line into L , . However, it does not satisfy (2) because II$(t - * ) l l L 2 = I I $ I I L 2 , so that the integrand of (2) is in this case a constant with respect to t . This counterexample has still another implication. It is a fact that any f E [ Q L 2 ; C] has a representation as a finite sum of derivatives of functions in L,; i.e.,

f

=

2hy,

hk E L , .

k= 1

(See Schwartz, 1966, p. 201.) This is no longer true for every f E [gL2; B], where B is an arbitrary Banach space. Indeed, if it were true, we could choose B = L , as above and then write, for any E 9,

c

* $ = 1 h p * If5 = hk * * ( k ) , where h k E L,(B). It can be shown that h k * $(k) E L,(B) so th a tf * I,$E L,(B). But we have already noted in the preceding paragraph that f * Ic/ 4 L,(B) = f

L,(L,) for a properly chosenf. Problem 7.2-2. Let h EL,(B), where B is a Banach space, and let Show that h * II/ E g L 2 ( B ) .

7.3.

I// E 9.

SCATTER-PASSIVITY

Let u be the input signal on a Hilbert port under the scattering formulism. If q is an ordinary function at some instant of time t , then Ilq(t)llH2is the instantaneous power injected by the input signal into the Hilbert port.

7.3.

SCATTER-PASSIVITY

129

Similarly, if 2B is the scattering operator and if 2Bq is also an ordinary function at the instant t, then ll(2Bq)(t)llH2is the instantaneous power carried out of the Hilbert port by the output signal. Therefore, the net power absorbed by the Hilbert port at time t is Ilq(t)llZ- II(2Bos)(t>l12.

(1)

In general, neither q nor ‘2Bq need be an ordinary function at t and indeed they may be singular distributions throughout some neighborhood of t. In this case, ( 1 ) will not possess a sense as the net power absorption. Nevertheless, our intent is to allow distributional inputs and outputs and at the same time make use of a passivity assumption. This is accomplished by first assuming that 2B is passive on a domain of ordinary functions [namely 9 ( H ) ] ,then developing certain representations for 2B, and finally extending 2B onto wider domains by means of those representations. (In this regard, see also the discussion at the beginning of Section 4.6.) Definition 7.3-1. Let 2B be an operator whose domain contains a set 3 c L , ( H ) . 2B is said to be scatter-semipassive on X or, alternatively, contractivefrom X into L , ( H ) if, for all q E X and for r a 2Bq, we have that r E L,(H) and

If, in addition, 3 = L2(H), then we simply say that 2B is contractive on LZW. A stronger condition than scatter-semipassivity is stated by the next definition, where the following notation is used. Given any T E R , we define the function 1, on R by l T ( t )= 1 for t IT and l T ( t )= 0 for t > T. If h is any function on R,the function tw l,(t)h(t) is denoted by 1,h. Definition 7.3-2. Let 2B be an operator whose domain contains a set 9 with the property that 1,q E L,(H) for every q E 9 and every T E R . 2B is said to be scatter-passive on g if, for all q E 9 and all T E R and for r A 2Bq, we have that 1 r E L 2 ( H )and

,

d -

m

If 2B is scatter-passive on X c L,(H), then it is also scatter-semipassive on X, as can be seen by letting T + 03. However, the converse is not true in general. For example, a pure predictor defined by (!Dq)(t)4 q(t x > 0, is scatter-semipassive but not scatter-passive on 9 ( H ) .

+ x), where

130

7.

THE SCATTERING FORMULISM

Lemma 7.3-1. Let 2B be a linear scatter-semipassive operator on .9(H). Then, ll3 is continuous from 9 ( H ) into L 2 ( H ) and therefore inlo [ 9 ;HI" as well. Moreover, 2B has a unique linear contractive extension onto L2(H). PROOF. The inequality (2) states in effect that, when 9 ( H ) is equipped with the topology induced by L,(H), 2B is continuous from 9 ( H ) into L2(H).But, the canonical injections of 9 ( H ) into L 2 ( H )and of L 2 ( H )into [ 9 ;HI" are both continuous, and this implies the first sentence. The second sentence follows from Appendix D5 and the fact that 9 ( H ) is dense in L2(H).0

Not only does the scatter-passivity of a linear operator imply continuity, it also implies causality. In fact, we have the following result originally pointed out by Wohlers and Beltrami (1965). Theorem 7.3-1. Let 2B be a linear operator on 9 ( H ) . Then, 2B is causal and scatter-semipassive on 9 ( H ) ifand only i f 2 B is scatter-passive on 9 ( H ) . PROOF. Let 2B be scatter-passive on 9 ( H ) and, as before, set r 2Bq, where q E 9 ( H ) . We have already noted that (2) can be obtained from ( 3 ) by taking T - t co,and therefore 2B is scatter-semipassive. Next, assume that 2B is not causal. This means that, for some T E R, we have q(t) = 0 for - 03 < t < T and r(t) # 0 for all t in some set of positive Lebesgue measure contained in (- 03, T ) . Then, (3) cannot hold, and this contradicts the scatter-passivity of $11.Hence, 2B must be causal on 9 ( H ) . Conversely, assume that '123 is causal and scatter-semipassive on 9 ( H ) . Given any T E R , choose an X E R such that T < X . Let 8 E d be real-valued and such that e(t) = 1 for - 00 < t 5 0, e(t) = 0 for 1 5 t < co, and 8 is monotonic decreasing for 0 < t < 1. Set

~ ( t ) l Tx,( t ) 4

e(-).Xt -- TT

Given q E 9 ( H ) , set g = !lB(lq) and r = 2Bq. By scatter-semipassivity, both g and r are members of L 2 ( H )and by causality, g = r almost everywhere on ( - 00, T ) . Thus, we may write

The second inequality is due to the scatter-semipassivity of $11.By choosing X sufficiently close to T, 1 ; IIlq1I2 dt can be made arbitrarily small because

This establishes (3) and thereby the scatter-passivity of 2B. 0

7.4.

BOUNDED* SCATTERING TRANSFORMS

131

Theorem 7.3-1 shows that the following two assumptions on an operator

2B are equivalent: (i) 2B is linear, causal, and scatter-semipassive on 9(H). (ii) 2B is linear and scatter-passive on 9 ( H ) . Under a postulational approach to realizability theory, (i) may be considered a better form for a hypothesis because causality and scatter-semipassivity are independent conditions (see Wohlers and Beltrami, 1965 or Zemanian, 1968b). However, statement (ii) recommends itself by virtue of its conciseness. Later on, the reader should bear in mind that causality is a consequence of (ii).

7.4. BOUNDED* SCATTEIUNG TRANSFORMS

Definition 7.4-1. A function S of the complex variable ( is said to be a bounded* mapping of H into H (or simply bounded*) if, on the half-plane C , 4 {(:Re > 0}, S is an [H; HI-valued analytic function such that II S(C)II[H ;H] 5 1.

Our aim in this section is to show that the Laplace transform of the unitimpulse response of a linear translation-invariant scatter-passive operator on 9 ( H ) is bounded*. The converse assertion (namely, every bounded* mapping is such a Laplace transform) will be established in the next section. Theorem 7.4-1. If the operator 2B is linear, translation-invariant, and scatter-semipassive on 9 ( H ) , then 2B = s * on 9 ( H ) , where s E [9,,(H); HI.

PROOF.By Lemma 7.3-1, $113is continuous from 9 ( H ) into [9; H]”. There-

= s *, where s E [ 9 ( H ) ;HI, according to Theorem 5.10-1. Let denote the norm for L,(H). The scatter-semipassivity of 2B implies that, for all $ E 9 ( H ) ,

fore,

11

*

[ILz

Lemma 7.2-2 now shows that s E [9,,(H); HI. 0 Theorem 7.4-2. If the operator 2B is linear, translation-invariant, and scatter-passive on 9 ( H ) , then its unit-impulse response s possesses a Laplace transform whose strip of definition contains the halfplane C , 4 {C: Re C > 0).

7. THE SCATTERING FORMULISM

132

PROOF. By Theorems 7.3-1 and 7.4-1, B3 = s *, where s E [9L2(H);HI, and B3 is causal. Hence, supp s c [0, 00) according to Theorem 5.1 1-1. We now

invoke Lemma 7.2-1 to conclude that s E [ Y ( O , and has a strip of definition containing C + . 0

00;

A ) ; B] so that 2 s exists

Lemma 7.4-1. Let a, b E R be such that b < 0 < a. Assume that s E [9L2(H); b(H), s * 4 exists and is a H ] and supp s c [O, a)).Then, for each 4 E Pa, smooth H-valued .function. Moreover, there exists a constant L > 0 and an integer I2 0 such that ~l(s* +)(t)ll I Le-b' max supllebr4(k)(t) 11. OSksl rER

(1)

PROOF.Since [9L2(H);HI c [9L,(H); HI, we have from Theorem 5.5-2 that s * 4 is a smooth H-valued function and (8

* 4)(0= M X ) , 40 - x)>.

-+

E be~ such that 1(x) = 1 for < x < 00 and 1(x)= 0 for -00 < x < - 1 . Let Bp LA sup, 1 l ( p ) ( xI.) Then, there exists a constant M > 0 and an

Let A

integer 12 0 such that Il(s * 4>(t>ll= II<s(x), X x M t - x)>Il 03

<M

max OSkSl p = O

(i)Bk-pe-b'

ebxd x -1

suplleb'4(P)(t)ll. roR

This implies (1). 0 Lemma 7.4-2. Let ?lB be a linear translation-invariant scatter-passive operator on 9 ( H ) and let q E Y c , d ( H )where , d < 0 < c. Then, for r a 'Dqand for every T E R, we have that

PROOF.Let a, b E R be such that d < b < 0 < a < c. Given anyq E Y cd(H), , c 9 ( H ) that tends to q in Ya,b(H). we can find a sequence {4j}T=1 Indeed, let 1 E 9 be such that 1(t) = 1 for I t I < 1 and A(t) = 0 for I t I > 2. We may write

7.4.

133

BOUNDED* SCATTERING TRANSFORMS

Now, Dk-”[A(t/j)= I ] is equal to zero for I t I I j ; and for I t J > j , it is bounded by a constant that does not depend on j . Moreover, for each p , Ka b(t)

- 0

IIKa,b(t)DPq(t)llIsupllKc,d(t)DPq(t)II

(4)

~,,d(t) as j - , 00. Consequently, the right-hand side of (3) converges uniformly to 0 on R. Thus, upon setting $ j ( t ) 4 A(t/j)q(t),we obtain the sequence we seek. Next, note that 2l3 = s *, where s satisfies the hypothesis of Lemma 7.4-1 according to Theorems 5.11-1, 7.3-1, and 7.4-1. Set $ j A 2134j = s * + j . Since supp s c [0, m), $ j E 9 + ( H ) . Also, set r 2Bq = s * q. By Theorem 5.5-2, r E I ( H ) . So, 11q1I2, llr112, I14j112,and Il$j112 are all continuous functions on R. Furthermore, we may invoke Lemma 7.4-1 and the fact that I ebr/Ka, b(t)I I1 to write Irl>j

rsR

111>j

e*‘IIr(t)- $j(t)lI I L max supllebrDk[q(t)- 4j(t)lII -,0,

j -, co. ( 5 )

Osksl IER

Now, given any T E R,consider

1

T J-m

T

[11q112 -

I I ~ I dI t~ -I J- m [ll4j1I2- II$~II~I dt

sm + ST I I ~ I I I I ~

I

IIqII 114 - 4jll dt

-m

-

’r

+

I

dt

+ J’- m I I -~

T

-m

114 - 4jll

I

IIdjII dt

T

II$jll

dt.

(6)

-=

Since b < 0 a and since $ j -,q in 9“. b(H), there exists a constant K not depending o n j such that Ilq(t)ll I Ke-br and I14j(t)II I Ke-b’. Also, sup ebrllq(t)- 4j(t)ll -,0,

j -, 00.

IeR

These facts imply that the first and second integrals on the right-hand side of (6) tend to zero as j + co. Similarly, by virtue of Lemma 7.4-1 and (9, Ilr(t)II, II$j(t)II, and ebrIlr(t)- $j(t)ll satisfy similar conditions, and, as a result, the third and fourth integrals on the right-hand side of (6) tend to zero as j - , co. Condition (3) now follows from the scatter-passivity of 2B on 9 ( H ) and the consequent nonnegativity of the second integral on the left-hand side of (6). 0 The main theorem of this section is the following.

Theorem 7.4-3. VlB is a linear translation-invariant scatter-passive operator = s *, where s E [QL2(H); on 9 ( H ) , then 2l3 has a convolution representation HI and supp s c [0, m). Moreover, the Laplace transform S of s exists with a strip of deJinition containing C + {t; E C : Re t; > 0) and is bounded*.

7.

134

THE SCATTERING FORMULISM

PROOF.We have already established everything except for the condition

c

IIS(c)ll[H:Hl I 1 for all E C+ . (See Theorems 5.11-1, 7.3-1, 7.4-1, and 7.4-2.) Let a E H and T ER. Choose z E R such that 7 > T. Also, let 1 E d be such that A(t) = 1 for -co < t < 7 and A(t) = 0 for z 1 < t < 00. For any E C+ , set q(t) = aertA(t).Thus, with -Re ( = d < 0, we have that q E Y cd, ( H )for every c E R. So, we may use Lemma 7.4-2 and ( 2 ) for r = 2Bq.

+

c

We now invoke Theorem 5.5-2 to write r(t) = (s * qNt) = M x ) ,q(t - x ) ) . For any fixed t < T , q(t - x ) = aer('-x)when x is restricted to a sufficiently small neighborhood of [0, 03). But supp s c [0, co), and therefore r(t) = (s(x), aec(t-x))= ec'S(c)a. Thus, J -

m

a,

Since this holds for all a E H and since the integral on the right-hand side is positive, this implies that IIS(c)IICH;Hl 1 for any E C+ . 0

c

7.5. THE REALIZABILITY OF BOUNDED* SCATTERING TRANSFORMS

We will now show that every bounded* mapping of H into H i s the Laplace transform of the unit-impulse response of a linear translation-invariant scatter-passive operator on 9 ( N ) . We start with two lemmas.

Lemma 7.5-1. Let 4 E ~ ( Hwith ) s u p p 4 c [0, co). Then,
( E C.

0

Through successive integrations by parts, this becomes = Ja,4(k)(t)c-ke-ctd t , 0

and (1) follows by estimating the last integral. 0

7.5.

REALIZABILITY OF BOUNDED* TRANSFORMS

135

Lemma 7.5-2. Let g be a continuous H-valued function on R such that I l g ( ' ) l l H E L , . Also, let a E H . Then,

This lemma is an immediate consequence of Appendix D15 and Note I1 of Section 1.4, which continues to hold for improper integrals. The following is the realizability theorem for bounded* scattering transforms. Theorem 7.5-1. Corresponding to each bounded* mapping S of H into H , there exists a unique convolution operator '2B = s * on [ 9 ( 0 ,co); HI such that 2s = S on C , . Moreover, s E [ 9 , , ( H ) ; HI and supp s c [0, co). Furthermore, '2B is a linear translation-invariant scatter-passive operator on 9 ( H ) . PROOF. We may invoke Theorem 6.5-1 to conclude that S is the Laplace transform of an s E [ 9 ( 0 ,co;H ) ; HI, that the strip of definition for 2s contains C, , and that supp s c [0, co). Hence, the convolution operator '2B 4 s *, which by the uniqueness theorem of the Laplace transform is uniquely determined by S, is a linear translation-invariant causal mapping on 9 ( H ) . Moreover, the domain of s * contains [ 9 ( 0 ,00); HI according to Problem 5.3-3. It remains for us to prove that s E [ 9 , , ( H ) ; HI and that s * is scatter-passive on 9 ( H ) . Let 4 E 9 ( H ) with supp 4 c [0, co). Then, s * 4 is Laplace-transformable, and by Theorem 6.3-2,

P ( s * f1l(5) = m-)F(C),

Moreover, for all [ E C, and for o

5 E c+

Im c,

where Co and C, are the constants indicated in (1). Set h = s * 4. Then, h E &(H),and supp h c [0, a).We now invoke Lemma 6.5-1 and the uniqueness theorem of the Laplace transformation (Theorem 6.4-2) to obtain the following result. For any fixed r~ Re 5 > 0 and for each fixed t E R, h(t)e-"'

= (1 /2n)

S(o

I O U

+ i o ) @ ( a + io)e'"' d o .

-m

In view of (2), IIh(t)e-"'II I (1/2n) Jm Q(o)do < co.

-

7.

136

THE SCATTERING FORMULISM

Since this holds for all o > 0 and since supp h c [0, a),it follows that, for any fixed a > 0, Ilh(t)e-"'ll E L , . Now, set H(c) S(c)cD(()for [ E C + and consider

rm

Ilh(t)e-"'IIZ dt

=

jm(h(t)e-"', (1/2n) -m

Jm -m

H(a + iw)eiw' dw dt

)

(3)

where again o > 0. By a judicious use of Lemma 7.5-2 and Fubini's theorem, we may first bring the integration on w outside the inner product in the righthand side of (3), then reverse the order of integration on w and ?, and finally bring the integration on t inside the inner product t o obtain Parseval's equation :

r*

1

00

Ilh(t)e-"'J12dt = (1/2n)

Since IIS([)ll I I for

-m

IIH(o + iw)l12do.

(4)

c E C , , the right-hand side of (4) is bounded by m

By (2) and Lesbesgue's theorem of dominated convergence, (5) tends t o

On the other hand, as a -+ 0 + , llh(t)e-"'112 increases monotonically t o the limit Ilh(t)l12 at each t . So, by the theorem of B. Levi (Williamson, 1 9 6 2 , ~62), . the left-hand side of (4) tends t o JZm Ilh(t)112dt. We conclude that

rm 11(s

* 4 ) ( t ) l l 2 dt

(1/2n)

jm

Il@(iw)ll2 dw


(7)

-03

for all 4 E 9 ( H ) with supp 4 c [0, co). But, since s * commutes with the shifting operator oT,

j;

Il(s m

* 9)(t)ll2dt < 03

for all 4 E 9 ( H ) . It now follows from Lemma 7.2-2 that s E [9,,(H); HI. To show that s * is scatter-passive on !3(H),we first observe that the middle term in (7) is equal to JTm I14(t)llzdt. [This is established just as was (4).] Thus, s * is scatter-semipassive on every 4 E 9 ( H ) with supp 4 c [0, a).But the same is true for all 9 E 9 ( H ) because again s * commutes with oT.The causality of s * in conjunction with Theorem 7.3-1 establishes the scatterpassivity of s *. 0 Theorems 7.4-3 and 7-5.1 summarize the basic realizability theory under the scattering formulism for arbitrary bounded* mappings of H into H .

7.6.

137

BOUNDED*-REAL SCATTERING TRANSFORMS

Actually, the hypothesis of Theorem 7.4-3 may be weakened somewhat by merely assuming that is linear, translation-invariant, and scatter-passive on the dense subset 9 0 H of 9 ( H ) (see Zemanian, 1970b). Another modification is to impose the passivity assumption on the admittance form of the energy integral and make use of the concepts of augmentation and solvability. This approach was devised by Youla et at. (1959) for the scalar case; see also Newcomb (1966). That this procedure can be extended to the Hilbert-port setting and is entirely equivalent to the method discussed in this chapter is shown by Zemanian (1970b, Section 7). An extension of the present theory to Hilbert ports that need be neither translation-invariant nor causal is given by Zemanian (1972b).

Problem 7.5-1. Let H and J be complex Hilbert spaces and let S and T be [ H ; J]-valued bounded analytic functions on C , . Assume that, for almost all w and as cr + 0 + , S(o + io)and T(a + iw) converge in the strong operator topology to S ( i o ) and T(io), respectively. Show that, if S(iw) = T(iw) for almost all o,then S = T o n C , . 7.6. BOUNDED*-REAL SCATTERING TRANSPORMS

A property of physical systems which we have not as yet considered is reality. That is, they map real signals into real signals. The meaning of a real signal in the context of Hilbert-space-valued functions has yet to be explained, and this is our first objective. The reality of a physical system is reflected in certain symmetries in its scattering operator and scattering transform, and an exposition of this fact is our second objective. We start with a real Hilbert space H, and generate a complex Hilbert space H by complexification. That is, the elements of H are defined to be a, ia2 , where a, and a, are arbitrary members of H , . Addition is defined as

+

(a,

+ ia,) + (b, + ib,)

(a,

+ b,) + i(a2 + b2),

and, for any complex number a = a, + ia, , where a,, a, E R, multiplication by a is defined by a(a, + ia2) A a l a l - a 2 a 2 ia,a, ia,a,.

+

+

+

H , becomes a subset of H when we identify a, i0 E H with the element a, E H , . Furthermore, the inner product (., .) on H , is extended onto H

through the definition

and as a result H becomes a complex Hilbert space. Throughout this section, it is understood that H is the complexification of H , .

7.

138

THE SCATTERING FORMULISM

A linear mapping of H into H is called real if it maps H , into H , . An arbitrary linear mapping Z of H into H has a unique decomposition Z = Z , + iZ,, where Z , and Z , are real linear mappings. Z , is called the realpart of Z , and Z , the imaginary part of Z . Given Z , we can obtain Z , and Z , as follows. Z maps any a, E H, into an element b , ib, of H . We set Z,a, 6 , and Z , a, b, . This defines Z , as a linear mapping of H , into H , , and it is extended onto H linearly:

+

Z,(a,

+ ia,) g Zlal + i Z , a , ,

a,, a2 E H , .

We proceed similarly for Z , . For no other decomposition of Z into the form Z = Fl iF, will both F, and F, be real mappings. The complex conjugate g f Z is by definition Z , - i Z , . Clearly, Z is a real member of [ H ; HI if and only if Z E [H,; H , ] . We now make a sweeping assertion. All the results we have so far developed in this book can be obtained in the context of real spaces. For example, instead of the basic testing-function space 9 ( H ) , we can use the space 9 ( H , ) of smooth H,-valued functions on R with compact supports. The members of [ 9 ( H , ) ;H , ] are real distributions, and f * with f~ [ 9 ( H r ) ;H , ] is a real convolution operator. Similarly, a real signal on a Hilbert port is an element of [ 9 ( R ) ;H , ] . Any f E [ 9 ( H r ) ;H , ] .has a natural extension onto 9 ( H ) defined by (f, 4 ) = (f, 4 , ) + i(.L 4 2 ) * where 4 = 41 + i 4 2 and 4 1 9 4 2 E g(Hr). Upon denoting this extension also by f,we havefE [ 9 ( H ) ;HI. In this way, any one of our standard spaces of real distributions is a subset of the corre; sponding space of distributions. For example, [B,,(H,);H,] c [ g L 2 ( H ) HI. Reality also has a meaning in the context of bounded* functions as follows.

+

z

Definition 7.6-2. A function S of the complex variable 5 is called bounded*real if S is bounded* and, for every o on the real positive axis (i.e., for a E R and a > 0 ) , the restriction of S(a) to H , is a member of [ H , , H , ] .

The next theorem is actually a corollary to Theorem 7.4-3. Theorem 7.6-1. If is a linear translation-inrariant scatter-passive operator on 9 ( H ) and if 21 maps Q(H,) into [ 9 ( R ) ;H,], then 2U = s *, where s E [ 9 , , ( H , ) ; H,], and S 2 s is bounded*-real. PROOF.By Theorem 7.4-3, 2B = s *, where s E [ 9 L z ( H ) ;HI, and S is bounded*. Choose a sequence {$j}j"=o c 9 ( R ) which tends to 6 in [ E ( R ) ;R ] . Then, for any a E H , and any 4 E 9 ( R ) , ('rO($j a), 4 ) E H , . But <m($ja)q

4 ) = (s(t)t

($j(x)a, $(t

+ XI>) 40) +

(

~

3

Therefore, (s, $a> E H , . But 9 ( R ) 0H , is total in 9 ( H r ) ,and so s E [ 9 ( H r ) ; H , ] . Hence, by the density of 9 ( H , ) in 9,,(H,), we have s E [ g L Z ( H r )H; , ] .

7.7.

139

LOSSLESS HILBERT PORTS

It now follows as in the complex case that s E [L?(O, 00); [H,; H,]] since supp s c [0, co) according to Theorem 7.4-3. Thus, for any a > 0, S(a) = ( s ( t ) ,e-"') E [H,; H,]. 0 As a corollary to Theorem 7.5-1, we have the following result. Theorem 7.6-2. If S is bounded*-real, then, in addition to the conclusion of Theorem 7.5-1, we haoe that s E [ 9 L z ( H r )H,] ; and s * maps 9 ( H , ) into [ g ( R ) ;Hr1. PROOF. We know from Theorem 7.5-1 that s E [ g L Z ( H )HI. ; The conclusion s E [ g L 2 ( H r )H,] ; will follow as in the preceding proof once we show that s

maps 9 ( H , ) into H,. This will also imply that s * maps g ( H , ) into [ 9 ( R ) ;H,]. Let 4 E 9 ( H r ) .According to the inversion formula for the Laplace transformation (Theorem 6.4-I), (s,

4)

=

lim w-m

(U/W

lW

--w

~ e c ' d w4,( t ) ) ,

where Re 4' > 0 and w = Im 4'. We have to show that ( s , 4) E H,. But this will be established when we prove that

Jyw~ ( w c '

dw E [ H r ; ~

r

1

(2)

for each w > 0 and t . Let a, b E H,. Then, (S(a)a,b ) is real for 0 real and positive. By the reflection principle and the decomposition S(c) = S , ( c ) + is,([),where S , , S2 E [H,; Hr1, we have ( S , t b , b) = (sl(Oa,b) and (S,(%)a,b) = - (S2(l)a,b) for E C , . Since this is so for every a, b E H,, S,(a + iw) and S,(a + iw) are respectively even and odd functions of w . We now obtain ( 2 ) by noting that the imaginary part of the integral there is the zero member of [H,; H,]. 0 Problem 7.6-2. Verify that the complexification of a real Hilbert space is a complex Hilbert space. Problem 7.6-2. Assume that 2J.I is a linear translation-invariant scatterpassive operator on 9 ( N , ) with range in [B(R);H,]. Show that 2J.I has a unique extension as a linear translation-invariant scatter-passive operator on %H). 7.7. LOSSLESS HILBERT PORTS

There are physical systems with the property that their net energy absorptions for all time from signals of compact support are always zero. Any electrical network consisting exclusively of a finite number of inductors and

7.

140

THE SCATTERING FORMULISM

capacitors has this property. Passive systems of this sort are usually called lossless. In regard to Hilbert ports, we shall use the following definition. Definition 7.7-1. Let (m be an operator whose domain contains a set (m is said to be lossless on X if 2J.l is scatter-passive on X and, for every q E X and r 4 (mq,

X c L,(H).

When X supplied with the topology induced by L,(H) is a normed linear space and when 2J.l is linear, condition (1) can be restated by saying that '2B is isometric from X into L,(H). It should be borne in mind, however, that losslessness requires in addition that 2J.l be scatter-passive on fa(H). The pure 4 , t < 0, is isometric from 9 ( H ) into predictor, defined by 4 ~ 0 ~ where L , ( H ) but not scatter-passive on 9 ( H ) an d hence not lossless on 9 ( H )according to our definition. It should also be noted that a linear operator 2J.l is lossless on 9 ( H ) if and only if (1) is satisfied and (113 is causal on 9 ( H ) ; this is a consequence of Theorem 7.3-1. When the Hilbert space H is separable, a lossless convolution operator can be characterized by the fact .that its scattering transform S(o io) is bounded* and, as o -+ 0 + , S(o i o ) converges for almost all w to an operatorS(io)suchthat IIS(iw)all = llall foralla E H.AnylinearoperatorT~[ H ;HI having this property (namely llTall = llall for all a E H ) is said to be isometric on H. As our first result along these lines, we have the following assertion.

+

+

Theorem 7.7-1. Let H be a (not necessarily separable) complex Hilbert space. Assume that S is an [ H ;HI-ralued analytic function on C+ such that, for all E C, , IIS(l)II I P( 1 (I), where P is a polynomial. Also, assume that, as o -+ 0 + and for almost all o,S(a + iw) -+ S(iw) in the strong operator topology, where S ( i o ) is a linear isometric operator on H. Set s = i?-'S (i.e., s is the unique member of [ 9 ( H ) ;HI whose Laplace transform coincides with S on C+). Then, s * is lossless on 9 ( H ) , and therefore S is bounded*.

c

PROOF. We first observe that, by Theorem 6.5-1, s E [ 9 ( H ) ;HI and supp s c [0, 00). Next, we let 4 E 9 ( H ) with supp 4 c [0, 00) and let @([) = 4(t)e-{' dt as before. @ is analytic on C. By the estimate of Lemma 7.5-1, we have, for 0 < a < 1,

IIS(a

+ iw)@(o+ io)l125 Q(w),

where Q E L , . Moreover, for any fixed o where S(o + io)converges strongly as o -+ 0 + , we may write IIS(a

+ io)@(o+ i o ) - S(io)@(iw)llIIIS(o + iw)II II@(o+ i o ) - @(io)II + II[S(a + iw) - S(io)]@(io)ll.

7.7.

141

LOSSLESS HILBERT PORTS

+

By the principle of uniform boundedness, IIS(a iw)lJ is bounded for 0 c a < 1. We can conclude that the left-hand side tends to zero as a -+ O + . Thus, by Lebesgue’s theorem of dominated convergence and Parseval’s equation,

1

1 “ lim IIS(a + iw)@(a + iw)I(’ dw a - ~ +27~ - m

1

1 IIS(iw)@(iw)11’ dw 271

=-

Moreover, Parseval’s equation [see (4) of Section 7.51 and the theorem of B. Levi show that the left-hand side is equal to lim a-O+

j Il(s * 4)(t)e-a‘112dt = s l l ( s * 4)(t)I12dt.

Thus, s * satisfies (1) for every q = 4 E 9 ( H ) with supp 4 c [0, co).The translation invariance of s * shows that s * satisfies (1) for all 9 E 9 ( H ) . Since supp s c [0, a),s * is causal and therefore lossless on 9 ( H ) . Theorem 7.4-3 now implies that S is bounded*. 0 Note. A small modification of the foregoing proof shows that, if we weaken the hypothesis of Theorem 7.7-1 by replacing the assumption that S(iw) is a linear isometric operator on H by the condition IlS(iwll I 1 for almost all w , then we can still conclude that S is bounded*. The next theorem is a sort of converse to Theorem 7.7-1. Its proof makes use of the following known fact. Let ff and J be separable complex Hilbert spaces. If F i s an [H; J]-valued function that is analytic and bounded on C , , then there is a set R c R whose complement R\R has measure zero such that, for each w E R and as a -+ O+, F(a + iw) converges in the strong operator topology. This is proven by Sz.-Nagy and Foias (1970, pp. 185-187) for [ H ; J]-valued bounded analytic functions on the unit disc {z E C: IzI < I } and strong convergence along radial lines. The present version can be obtained by mapping the unit disc into C + through = ( I + z)(l - z)-’. The radial lines becomes circles centered on the imaginary axis and passing through 5 = 1. That strong convergence for C(c) A F(z) also occurs along horizontal lines in the [ plane follows from the inequality

c

IIG(0

+ iw) - C(0 + iq)II I 4 ( 1 w - ql/a)SUP

SEC+

IIG(i)ll,

0 <0 <

+

(2)

which is a result of Cauchy’s integral formula. Theorem 7.7-2. Let H be a separable complex Hilbert space. Assume that the convolution operator s *, where s E [ 9 ( H ) ;HI, is lossless on 9 ( H ) . Then,

7. THE SCATTERING

142

FORMULISM

+

S A Ss is bounded*, and,for almost all w and as a -+ 0 , S(o + iw) conoerges in the strong operator topology to a linear isometric operator S(iw) on H . PROOF.By the definition of losslessness, s * is scatter-passive on 9 ( H ) ,and consequently S is bounded* according to Theorem 7.4-3 and the linearity and translation invariance of convolution operators. We again invoke Parseval's equation to write, for any 4 E 9 ( H ) with supp 4 c [0, co),

I

Il(s

* 4)(t)e-url12dt = (1/2n) IllS(a + iw)@(a+ iw)IIz dw.

As before, the left-hand side tends to and Parseval's equation, is equal to

II(s

(3)

* 4)(t)l12dt, which, by losslessness

( l i 2 4 I Il@(iw)l12h. On the other hand, since IIS(LJ11 5 1 for l E C, , there exists a set R c R with R\R of measure zero such that, as a 0 + , S(a iw) -+ S(iw) E [ H ;HI strongly for every w E R. It follows readily that IIS(iw)ll I1 for all w E IR. We can now employ the principle of uniform boundedness and Lebesgue's theorem of dominated convergence as in the preceding proof to conclude that the right-hand side of (3) converges as a + O + . Altogether, then,

+

-+

I

II@(iw)l12dw

=

s

IIS(iw)@(iw)112dw.

(4)

But ( S a T 4 ) ( i w= ) e-iuT@(iw), and consequently (4) holds for all 4 E 9 ( H ) and not merely for those 4 with supp 4 c [0, a). We shall use (4) to show that IIS(iw)all = llall for every a E H and almost all w , thereby completing the proof. To this end, let f,,d(w) = 1 for c < w < d and f , , d ( W ) = 0 otherwise. We use the classical fact that the Fourier transformation 8 is an automorphism on L, . Choose a sequence {e,}i",, c 9 that converges in L , to the inverse Fourier transform of f , , d . We have that O,(iw) (SOj)(iw)= (8O,)(o). Thus, O j ( i .)a f c V d ( * )ina L,(H). Observe that -+

j IIS(iw)[fc,d(w)a- Oj(iw)a]l12dw I jllfc,d(w)a- Oj(io)al12dw

--*

0.

Thus, upon setting @(iw) = Oj(iw)ain (4) and taking the limit, we get

Since this holds for every c and d , we have llall all w . 0

=

IIS(iw)all for all a and almost

A unitary operator on H i s a linear isometric operator that maps H onto H. When H i s finite-dimensional, every linear isometric operator on H i s unitary.

7.8.

THE LOSSLESS HILBERT PORT

143

However, this need not be the case when H is infinite-dimensional. This is reflected in the fact that, when s * is lossless on 9 ( H ) , S(iw) is isometric almost everywhere but not necessarily unitary. The following is an illustration of this. Example 7.7-2. Let {e,}i”,, be an orthonormal basis in the separable, infinite-dimensional Hilbert space H. Every a E H has the unique expansion a = crjej, where ccj E C. Moreover, lla112 = I J ~ l ~ 1Define 1~. an operator F on a by

1

1

m

Fa = ~ u j e j ,. , j= 1

F is linear and isometric on H but not unitary. We now define an operator

+

‘223 on 9 ( H ) by (‘223+)(t) F[+(t)],where E 9 ( H ) . Clearly, ‘223 = s *, where s = Fa. Therefore, S(C) 4 (Ps)(C) = F for every [. Thus, S(io) is isometric but not unitary for every o. 0

Problem 7.7-2. Derive (2) and prove the assertion about the strong convergence of C(C) along horizontal lines being a consequence of strong convergence along certain circles. 7.8. THE LOSSLESS HILBERT n-PORT

The Hilbert n-port can be viewed as a mathematical model for a system having n places of access through which energy can be injected or extracted. A cavity resonator having n waveguides connected to it is such a system. We define the Hilbert n-port as follows. Let H i , where j = 1 , . . . , n, be complex Hilbert spaces with the inner products (., * ) j and norms I I . I l j . We shall drop the subscription j on the inner product and norm notations when it is clear which Hilbert space Hi is meant. Let H 4 H I x . * * x H, be the Cartesian product of the Hi and assign to H the product topology. H is also a Hilbert space whose inner product is defined on any a = { a j } and b = { b j }in H by ( a ,

a )

( a j , bj).

(a, b) 4 j= 1

Thus, the norm for H satisfies n

Every F E [ H ; HI has a unique n x n matrix representation [Fkj],where FkjE [ H i ; Hk], and conversely every such n x n matrix defines a member of

144

7.

THE SCATTERING FORMULISM

[H; HI when the customary rules for matrix multiplication are followed. Upon denoting the adjoint of an operator with a prime, we have F ‘ = [Fkj]’ = [Fjk]; that is, we obtain the adjoint of [Fkj] by interchanging the rows and columns and then taking the adjoint of each element. Moreover, if F ( . ) is an [H; HI-valued analytic function on some open set R c C , then we may write, for any a, b E H ,

Since the weak analyticity of an operator-valued function is equivalent to its analyticity (Theorem l.7-1), F ( * ) is analytic on if and only if Fkj(’) is analytic on R for every k and j . With H = H, x * * x H,, given, we define a Hilbert n-port as a Banach system that relates two H-valued distributions q and r that are complementary in the following sense. Whenever q and r are ordinary functions, 11q(t)l12 - Ilr(t)I(’ is the instantaneous net real power entering the Hilbert n-port at the instant t . We obtain the ordinary n-port of electrical network theory upon setting each Hi = C . The mapping q H r is the scattering operator 2B. According to our prior results, 2B is a linear translation-invariant scatter-passive operator on 9 ( H ) if and only if I1u = s *, where S 4 i 2 s is bounded*. Upon setting v = q + r and u = q - r, we define the admittance operator % as the mapping v Hu. Let H a n d J be complex Hilbert spaces and let T be an [H; J]-valued function on some set Z. We shall say that T is bounded by 1 on Z if IlT(c)ll 5 1 for all ( E Z. The matrix representation [skj] of any bounded* mapping S of H into H has the following property: Every s k j is bounded by 1 on C , , and therefore each s k k is a bounded* mapping of Hk into H k . T o demonstrate this, we write

Upon setting all aj = 0 except when j = I, we obtain k

~ ~ s k l ( ~ ) a5l ~Ilall12? ~2

which implies that llskl(c)11 5 1. Our purpose in this section is to examine the properties of lossless Hilbert n-ports that have some additional special properties. The results to be given here are taken from the thesis of D’Amato (1971). They are similar to certain known facts concerning the ordinary lossless n-port (Carlin and Giordano, 1964), but there are also some notable differences. Throughout the rest of this section, we shall assume that the complex Hilbert spaces H , J , and H, are separable. Let T be an [H; J]-valued bounded

’

7.8.

THE LOSSLESS HILBERT

n-PORT

145

analytic function on C , . We have already noted in Section 7.7 that, as a -+ O + and for almost all w , T(a + io)tends to a limit T(iw) in the strong operator topology. In the following, whenever we write T(io),it is understood that T(iw) is the strong limit of T(o + iw) as a + O + for those o at which the limit exists. As was indicated in Problem 7.5-1, the boundary values T ( i o ) uniquely determine T o n C , . We shall call a scattering transform S lossless if s * is lossless on 9 ( H ) , where s E [ 9 ( H ) ;HI and S = 2s. By Theorems 7.7-1 and 7.7-2, S is lossless if and only if S is bounded* and S(iw) is isometric for almost all o.The isometry of S(io) is equivalent to the condition S’(io)S(io) = I , where I is the identity operator on H a n d S’(io) [S(io)]‘.I n terms ofthe matrix representa= [I,,], where l k j = 0 if k # j tion S = [Skj], this condition reads [Sjk][&j] and I,, 4 I, is the identity operator on Hi.Upon expansion, we get

+

S;k(iw)Sl,( io)

* * *

+ S;,(io)S,,,(io)= 0,

k #j

(1)

and

Sij(io)Slj(io) + * . . + Sij(io)Snj(io) = 1,

(2)

for almost all o.Thus, [S,,]is lossless if and only if it is bounded* and satisfies (1) and (2) for all k and j . The scattering transform of a Hilbert n-port is called reciprocalif its matrix representation [Skj] satisfies Ski = Sjk for all k an d j. The physical significance of this is that the transmission between any two ports does not depend upon the direction of the transmission. Many systems possess this property. It is implicit in the definition of reciprocality that the Hilbert spaces Hicomprising H = H, x * * * x H,, are all the same; that is, Hi= H , for some Hilbert space H O

.

If S = [ & j ] is a reciprocal lossless scattering transform of a Hilbert 2-port, then (2) yields

Sil(io)Sll(io) + Si1(io)S21(io) = I, and

Sil(io)Szl(io) + Si2(io)Sz2(io) = I,, where I, is the identity operator on Ho . Consequently,

S;,(iw)S,l(io) = S;z(iw)S22(io)

(3)

or equivalently ~ ~ S l l ( i o=) aIISzz(io)all ~~ for all a E H , and almost all w . To get a physical interpretation of (3), let 4 = {41,0} E 9 ( H o x H,) and expand the lossless condition

J I14(t>l12dt = J Il(s * 4)(O1l2dt?

7.

146

where s = [ s k j ] and s,, obtain

THE SCATTERING FORMULISM

= s,,,

through the usual matrix manipulations. We

As we have seen in the preceding section, this is equivalent to

The left-hand side represents the energy impressed upon the first port for all time. The first term on the right-hand side is the reflected energy at that port, and the last term is the energy transmitted from the first port to the second port, again for all time. A similar equation holds for the second port, and so we conclude that (3) has the following significance. Both ports of a Hilbert 2-port, whose scattering transform is reciprocal and lossless, behave the same way so far as total energy reflection is concerned. This is a classical result for ordinary 2-ports (Carlin and Giordano, 1964, p. 276). Another (albeit less common) property of Hilbert n-ports is the following. The scattering transform [Skj]of a Hilbert n-port is said to be matched at the ith port if S j j = 0. This means that thejth port does not reflect any energy. As an immediate consequence of our foregoing equations, we have the following extension of another classical result. If the scattering transform [ & j ] of a Hilbert 2-port is reciprocal, lossless, and matched at the first port, then it is also matched at the second port, and in addition S , , is lossless. For ordinary 2-ports, reciprocality is not needed in order for S , , = 0 to imply S , , = 0 (Carlin and Giordano, 1964, p. 379). However, when Ho is infinite-dimensional, reciprocality is needed, as is shown by the following example.

Example 7.8-2. We shall construct a lossless scattering transform [ & j ] for a Hilbert 2-port such that S , , = 0, S,, # 0, and S , , # S , , . Let H = H , x H , and let {e,},'", be an orthonormal basis for H , . An arbitrary a E H , has the unique expansion

,

m

a=

Caiei,

i= 1

a, EC.

We define S , , , S , , , and S,, as constant operators not depending on [, through the following equations: m

7.8.

THE LOSSLESS HILBERT n-PORT

It follows readily that S ; ,

= S , , , S ; , = S,,

147

, whereas

m

1ui+l e i . i= S ; , S12+ S;, S22= I , , S;,u =

1

Consequently, S ; , S 2 , = I , , 0. Thus, ( 1 ) and ( 2 ) are satisfied, and, for

and S;, S 2 , = S;,S,, =

we have that S'S = I , where I is the identity operator on H . Thus, S is isometric, and llSll = 1. Since S does not depend on (, we conclude that S is bounded* and therefore lossless. 0 We now turn to the Hilbert 3-port. Another standard result is that there does not exist an ordinary 3-port whose scattering transform is reciprocal, lossless, and matched at all three ports (Carlin and Giordano, 1964, p. 278). This restriction is no longer in force when H = H , x H , x Ho and H , is infinite-dimensional. However, we can say the following. If S is a reciprocal lossless scattering transform for a Hilbert 3-port and is matched at all three ports, then H , is infinite-dimensional and the following conditions are satisfied for almost all o:

S;,(iw)S,,(io)= S;3(i~)S13(i~) = S;3(io)S23(iw) = +I,,

(4)

S;z(i~)S13(i0) = S;z(i~)S,3(iw) = Si3(io)S23(io) = 0.

(5)

These equations are obtained by substituting the conditions S,, = S 2 , = S,, = 0 and I j = I , in ( 1 ) and ( 2 ) and solving. To show that H , is infinitedimensional, we first observe from (4) that both J2Sl,(io) and ,/2Sl3(io> are isometric on Ho for almost all w . Now, let 9 [ T ]denote the range of an operator T. Also, let . " T I be the null set of T, that is, the set of all points a for which Tu = 0. The isometry of JZS13(iw)implies that 9 [ S 1 3 ( i w )#] {0}, and (5) indicates that X [ S ; , ( i o ) ] 2 9 [ S 1 3 ( i w ) ]again , for almost all o. But N [ S ; , ( i w ) ]is equal to the orthogonal complement of W[S,,(iw)],that is, to the set of all ~ E H for , which (a,b) = 0 for all b~.@[S,,(iw)] (see Berberian [ l ] ,p. 133). We conclude that there exists at least one w for which 9 [ S 1 2 ( i o )#] Ho , whereas ,/2S,,(iw) is an isometry. This can happen only when H , is infinite-dimensional (Berberian, 1961, p. 146). Example 7.8-2. Here is an example of a reciprocal lossless scattering transform S = [ S k j ]for a Hilbert 3-port that is matched at all three ports. As before, let H = If, x H , x H , , where H , is infinite-dimensional, and let { e i } z o be an orthonormal basis for H , . We take S to be constant on the

148

7. THE SCATTERING

FORMULISM

plane and set S,, = S22= S3, = 0. The other elements of S defined on any a = Egoa i e i by S,,a

=-

l

JZ

1

=

[skj]

are

m Caie3i, i=o

m

It follows that

Therefore, S'S = I . Since S is constant on the ( plane, S is reciprocal, lossless, and matched at all three ports. 0 Problem 7.8-1. Let H = Ho x . . . x H , , where Ho appears n times. Assume that S = [ & j ] is a bounded* mapping of H into H. Also, assume that Sif is a = 0 for lossless scattering transform for one fixed pair i and 1. Show that all k # i.

Chapter 8

The Admittance Formulism

8.1. INTRODUCTION

In this final chapter, we discuss the admittance formulism for the Hilbert port. Its distinguishing characteristic is that the net real energy absorbed by the Hilbert port over the time interval I is e(1) = Re

1

(%zv(t), u(r))

dt,

I

where is the admittance operator. The passivity of ’J1 is reflected by the condition e ( l T )2 0 for every time interval of the form { t : -a < t S T } , where T < 00. The admittance formulism is inherently more complicated than the scattering formulism. For example, as was indicated in Lemma 7.3-1, a linear scatter-passive operator on 9 ( H ) has a continuous extension onto L,(H), which is contractive. The use of this fact in conjunction with certain forms of Parseval’s equation led to the realizability theorems of Sections 7.4 149

150

8.

THE ADMITTANCE FORMULISM

and 7.5. On the other hand, under the admittance formulism, a linear passive operator on 9 ( H ) need not have a continuous extension onto L,(H). An illustration of this is given in the next section. As another example, recall that the causality and scatter-semipassivity of a linear scattering operator is equivalent to its scatter-passivity. In contrast to this, the connection between semipassivity and passivity for a linear admittance operator is more complicated and involves a certain restriction on the singular behavior of the admittance operator; see Theorem 8.13- 1. Most of this chapter is based on the work of Hackenbroch (1968, 1969), which is an extension in several ways of the classical paper of Konig and Meixner (1958) dealing with systems having complex-valued signals. In addition, some of these results are reworked in a distributional context to obtain the conclusions of Zemanian (1903, 1970a). In this chapter, the integration theory of Chapter 2 is used in an essential way. Moreover, we assume throughout that t E R and 9 = g R L . 8.2. PASSIVITY

We start with the definition of semipassivity, which is a weaker condition than passivity. In the following, L','"(H) denotes the set of all H-valued functions on R which, for every compact interval I c R, are members of & ( I , C; p ; H ) , where C is the a-algebra of Bore1 subsets of I and 11 is Lebesgue measure. L , is the special case obtained by setting I = ( - 00, co)and H = C . Definition 8.2-1. Let % be an operator whose domain contains a set c L p ( H ) . 91 is said to be semipassive on X if % maps X into L p ( H ) and if, for every u E 3 and for u A %v, we have that

X

(4%4 9 ) E

L,

(1)

and

In the next definition, 1, is the function defined in Section 7.3; namely IT(t) = 1 for t I T a n d 1, = 0 for t > T. Definition 8.2-2. Let % be an operator whose domain contains a set c L',O'(H). % is said to be passive on X if % maps X into L p ( H ) and if, for every u E 57 and every T E R and for u A 91v, we have that

I(T

(49?49)1T(.)

E

L,

(3)

8.2.

PASSIVITY

151

and

Whenever X is a set of continuous H-valued functions on R with compact supports [in particular, when E = 9 ( H ) ] , conditions ( I ) and (3) are automatically satisfied by virtue of the assumption that % maps X into L P ( H ) . For, (u(.), u(.)) is clearly measurable (see Appendix G9). Moreover, if K is a compact interval containing supp u, then

Obviously, the passivity of % on 9 ( H ) implies its semipassivity on 9 ( H ) . However, the converse is not true in general even when % is causal. Indeed, let u 4?i %u 4 -dl)for all u E 9 ( H ) . Then, upon integrating by parts, we see that Re

I'

-m

( u ( t ) , 4t)) dt = -tllu(T)1I2.

As T - r co, the right-hand side tends to zero, and so % is semipassive on 9 ( H ) . However, '3 is certainly not passive on 9 ( H ) . In Section 8.13, we will determine the precise conditions under which passivity and semipassivity are equivalent for a convolution operator. On the other hand, %: VH u(') is clearly passive on 9 ( H ) . This is an example of a linear passive operator on 9(H) that does not have a continuous extension onto L,(H). Another point should be made here. The admittance variables u and 11 are related to the scattering variables q and r by v=q+r,

u=q-r.

Upon substituting these equations into (4), we obtain the energy condition for scatter-passive operators: T

J-

[114(t)1I2 m

Ilr(t)1l21dt 2 0.

However, the assumption that the admittance operator U H U is passive on 9 ( H ) is not quite the same as the assumption that the scattering operator q H r is scatter-passive on 9 ( H ) . This is because q is not in general a member of 9 ( H ) when u is a member of 9 ( H ) , and conversely. We now present a result of Youla et al. (1959). Linearity and passivity imply causality. The analog to this under the scattering formulism is contained in Theorem 7.3-1. Under the admittance formulism, we have the following.

8. THE ADMITTANCE FORMULISM

152

Theorem 8.21. Let 92 be a linear passive operator on 9 ( H ) . Then, 92 is causal on 9 ( H ) . PROOF. Let u, ul E 9 ( H ) . Set u = 92u and u1 = 92ul. By the definition of passivity, u, u1 E L ~ ( H )Given . an arbitrary T ER, assume that u(t) = 0 for t < T. We shall show that u(t) = 0 for almost all t < T, proving thereby that 92 is causal. Let u be any real number and set u2 4 u1 + uu and u2 A u1 + uu. By the linearity of 92, u2 = 92u2. Therefore, by passivity, T

Re

j-( u 2 , u2) dt 2 0. m

Since u2(t) = ul(t) for t < T , this can be rewritten as T

T

( u l , ul) d t

ReL, But u is arbitrary, and hence

+ u Re j-m (u, u l ) dt 2 0.

T

Re

1-

(u, ul) dt = 0.

9

Upon replacing u1 by iul, we see that the imaginary part of the integral is also equal to zero. Therefore, T

J-,Cu,

v1)

dt = 0.

(5)

Now, let K be any compact interval contained in (- co, TI. By Schwarz's inequality,

Since 9 ( H ) is dense in L2(H), it follows easily that SK I(u - u1112 dt can be made arbitrarily small by choosing an appropriate u1 E g K ( H ) . In view of (5), this implies that jR llul12 dt = 0. Since K is arbitrary, we conclude that u(t) = 0 for almost all t E (- 00, TI. 0 Problem 8.2-1. Let H be the complexification of a real Hilbert space H,. Assume that 92 is a real linear mapping on 9 ( H ) . Thus, 92 maps 9(HJ into [ 9 ( R ) ;H,]. Show that, if 92 is passive on 9(Hr), then it is also passive on 9(H). Problem 8.2-2. Let 92 = f., where f ( t , x) = h(t)h(x), h E Lp(R), and h is not a function that equals zero almost everywhere. Show that 'illis a linear

8.3.

LINEARITY, SEMIPASSIVITY IMPLY CONTINUITY

153

semipassive operator on 9 but is not passive on 9. Also, show that W is not causal on 9 if supp h intersects the open interval (- o3,O). What must h be if 92 is to be translation-invariant?

8.3. LLNEARITY AND SEMIPASSIVITY IMPLY CONTINUITY

A remarkable result of Konig (1959) is that linearity and passivity imply continuity. [See also Hackenbroch, 1969 and Hackenbroch (to be published).] In the present context, we can state the following. Theorem 8.3-1. Let W be a linear semipassive mapping on 9 ( H ) . Then, W is continuousfrom 9 ( H ) into [9;HI. PROOF.

h t

Then, b is a sesquilinear form on 9 ( H ) , and Re b(4, 4) 2 0 by semipassivity. We first prove that, for any compact interval K c R,8 is continuous on gK(H)x 9,(H) supplied with the product topology. Upon applying the Schwarz inequality (Appendix C15) to (I), we see that

I 2x49 $1 I

I 4(4)11 $llLz 9

49 $ E ~ K ( H ) ,

(2)

-

where 11 IJL2 is the norm for L,(H) and q is a function from 9,(H) into R such that q($) > 0 for all 4 E 9#). Now, consider

8(49 $1 a8,(4, $1 4 “(4, $1 + 5G31. By virtue of the semipassivity of W, 8 is a positive sesquilinear form on 9,(H) x 9 , ( H ) . The Schwarz inequality (Appendix A6) yields

since

$1 I I [8(4, 4)8($9 $)11/2. I 4) I Is(+,$1I , we get I $1 I I I W$,$1 I + m 4 , 4)8($, $ ) P 2 I f w 4 9

W($)Z(4),

(3)

154

8. THE ADMITTANCE FORMULISM

Now, for every 4 E g K ( H )other than the zero function, the mapping

is continuous and linear from g K ( H ) into C. Thus, f4 E [ g K ( H ) ;C]. In view of (3), the collection of all suchf4 is a bounded set in [ g K ( H ) ;C]". Therefore, that collection is equicontinuous on g K ( H ) (Appendix D9). As a result of this and the continuity of 11 ]IL, on g K ( H ) , we can find a continuous seminorm p on g , ( H ) such that

Iw4,$1 I

5 P($)Z(4)

(6) $ E g K ( H ) . Combining this with

and simultaneously 1) $[IL2 5 p($) for all 4, (5), we now obtain Z(4)/P(4) 5 1 + [z(4>/P(4>11'2. Consequently, there exists a constant M > 0 such that Z ( 4 ) I Mp(4). By (6) again, I w49 $11 5 M P ( 4 ) P ( $ ) , (7) which verifies the continuity of 23 on g K ( H ) x g , ( H ) . Next, set JI = Ba, where 0 E g Kand a E H.By (7),

so,

II(W @llH =

SUP

llall = 1

I((W4, (0,a l l I SUP

llall = 1

MP(4)P(ae).

Let 0 be a bounded set in 9.Then, 0 is a bounded set in g N for some compact interval N. Moreover, given any other compact interval J , we can choose K to contain N and J. Then, p will be a continuous seminorm on g , ( H ) , and 0 will be a bounded set in g K .Consequently,for every 4 E g , ( H ) , SUPll(~4,OllH I PP(4),

eEe

where P is a constant. This proves that '9l is continuous from g , ( H ) into [g; HI. Since J can be chosen arbitrarily, '3 is continuous from 9(H)into [9;Hl. 0 Problem 8.3-1. Assign to L?(H) the topology generated by the collection

{q.}g of seminorms defined by Wk(f)

=

I

Kk

Ilf(t)llH

dt,

.fEL:"'(H)~

8.4.

THE FOURIER TRANSFORMATION ON

Y(H)

155

where {&};= is a nested closed cover of R. Then, Lp(H) becomes a Frkchet space. Use the closed-graph theorem (Appendix DlO) to prove that every linear semipassive mapping on 9 ( ~ is continuous ) from 9 ( ~ into ) LP(H).

8.4. THE FOURIER TRANSFORMATION ON Y ( H )

Subsequently, we shall need a variety of results concerning the Fourier transformation 8. We gather them in this section. The Fourier transform of any 4 E Y ( H ) is denoted by $ 4 84 and defined by

Some differentiations under the integral sign and integrations by parts show that

1

m

(io>~$(~)(o> = e-'mrD?[( - itlk+(t)l tit, -m

and this implies that 8 is a continuous linear mapping of Y ( H )into Y ( H ) . The same argument as that which holds for complex-valued functions (Zemanian, 1965, Section 7.2) establishes the following inversion formula for 8: m

+(t) = ( ~ - 1 $ ) ( t ) = (1/21t)

-m

(2)

$(w)e'a' dw.

Here, 3-l denotes the inverse Fourier transfbrmation. By the same manipulations as above, we see that S-', too, is a continuous linear mapping of Y ( H ) into Y ( H ) . It also follows that 3 must be a bijection. Thus, 8 is an automorphism on Y ( H ) . We take note of the following version of Parseval's equation: 2n J m (4
-m

($<w>, $W)dw,

4 9

+

E Y(H).

(3)

This can be obtained by using Fubini's theorem and Lemma 7.5-2. We now present three lemmas concerning complex-valued functions.

Lemma 8.41. Let 9 * 9 denote the set of allfunctions 4 with the representation q5 = 9 * $, where 8, $ E 9. Then, 9 * 9 is dense in 9.

156

8. THE ADMITTANCE FORMULISM

This is established by choosing a sequence {8k}z=l 8, dt = 1, and supp 8, c { t : 1 t I 5 k-'} and then showing that 8, * 4 +

x

c 9 such that 8, 2 0,

in 9.

Lemma 8.4-2. Let I a B(9). Also, let I * E denote the set of all functions with the representation x = #$, where 8, $ E 9.Then, I * 9 'is dense in 9.

PROOF.Choose any $ E Y and any Y-neighborhood fi of $. Then, B-'$ E Y, and R 6 S-'(fi) is an Y-neighborhood of 4. By the density of 9 in Y, there exists a E 9 n h. But, since the topology of 9 is stronger than that induced on it by Y and since h is also an Y-neighborhood of c, there exists a %neighborhood of ( such that E c h. By the preceding lemma, wecan find a 8 * $ E 9 * 9 in Z. Thus, 8 * $ E R. But B(8 * $) = #$. Therefore, #$ E 6.0

4

Lemma 8.4-3. Given any bounded Bore1 set E c R, there exists a function 8E9suchthat 2 1forallqEE.

PROOF. We can choose a compact interval K 3 E and a function E such that $(q) > 2 for all q E K. Therefore, $ E Y. We can also choose a sequence {$k}?= c 9 that converges in Y to $. Therefore, $I, -+ $ in Y . This implies that $k tends to $ uniformly on K. Thus, the sequence ( 4 k ) contains an element 8 with the required properties. 0 We finally take note of a standard result concerning complex-valued distributions, namely the Bochner-Schwartz theorem. In the following, 6 denotes the collection of all Bore1 subsets of R. A temperedpositive measure p is a positive measure on 6 such that (1 t 2 ) - p E Ll(R, 6; p ; C) for some integerp. (See Appendix G, Sections G4, G10, and G13.) On the other hand, f E [9; C] is called positive-definite if, for every 4 E 9,

+

Theorem 8.4-1. f E [g; C ] is positive-definite if and only if there exists a temperedpositive measure p such that, for every 4 E 9,

Proofs of this theorem are given by Schwartz (1966, pp. 276-277) and Gelfand and Vilenkin (1964, Section 3.3). According to the standard definition of the Fourier transformation on [CB;C] (see, for example, Zemanian,

8.5. LOCAL MAPPINGS

157

1965, Section 7.8), (4) means that f is the Fourier transform of the distribution generated by p. Later on, we shall meet a tempered complex measure. This is a a-finite complex measure v (see Appendix G12) for which

J-r.tldlvl (1 + t 2 ) - p converges as k --t co if p is chosen sufficiently large.

8.5. LOCAL MAPPINGS

Let W be a linear translation-invariant semipassive mapping of 9 ( H ) into [9; HI. By Theorem 8.3-1, W is also continuous. Therefore, 92 = y *, where y E [ 9 ( H ) ; HI, according to Theorem 5.10-1. It now follows from Theorem 5.5-1 that W is a continuous linear mapping of 9 ( H ) into b(H). We define the sesquilinear mappings 8 and b on 9 ( H ) x 9 ( H ) as before. For instance,

w4,$1 %l(4, *)

Srn(W4(0, W )

dt,

-m

4,

*

E9 w .

(1)

By the proof of Theorem 8.3-1, 8 is a continuous sesquilinear form on gK(H)x 9#) for every compact interval K , and therefore so too is 8 because

B(49 $1 a @9?(49*)

f

i&J f m

-m

w 4 , $1 + (4, %*)I

dt.

(2)

Another result from Hackenbroch (1969) is a characterization of those causal operators W for which b = 0. One property such operators have is that they respond only to present values of input signals, but not to past values nor, by causality, to future values. A precise definition of this property is the following.

Definition 8.5-1. A mapping W of 9 ( H ) into [9; HI is said to be local if supp % ' $ c supp 4 for all 4 E 9 ( H ) . For linear mappings, local operators are the same thing as the so-called memoryless operators of systems theory (Willems, 1971, p. 14). Theorem 8.5-1. Let W be a linear translation-invariant causal mapping of g ( H ) into [9;HI andsuch that b = 0. Then, W is local. Note. The condition b = 0 implies that W is semipassive on 9 ( H ) .

8.

158

THE ADMITTANCE FORMULISM

PROOF. That W is local is equivalent to the following assertion. If 4 E 9(H) vanishes on a neighborhood A of T ER, then W 4 also vanishes on A. Now, 4 can be written in the form 4 = 4, + &, where 4,, 42E 9(H), supp 41 c [t,, t 2 ] ,and supp 4zc [ f 3 , t,] for certain t,, t 2 , t 3 , t, E R with t, < t z < T < t 3 < t 4 . So we need merely prove that, if 4 E 9(H) and supp 4 c [c, d ] , then supp %q!~ c [c, d]. Since W is causal by hypothesis, W+ vanishes for t < c. Again by hypothesis, @(4,J / )= 0 for all E 9(H),so that

+

J(W4, J/) dt = - J(4, WJ/) dt.

(3)

Choose J/ such that J/(t)= 0 for t < d. Therefore, (W+)(t)= 0 for t < d by causality. Hence, the right-hand side of (3) is equal to zero. But the behavior of J/ for t > d is unrestricted, and we know that 8 4 E b ( H ) . Therefore, (W+)(t) = 0 for t > d as well. 0 In order to prove Hackenbroch's characterization of W (see Theorem 8.5-2), we shall need the following result.

Lemma 8.5-1. Let A and B . be complex Banach spaces. Assume that c [ A ;B],'where Lk # 0 for every k. Then, there exists an a E A such that Lk a # 0for every k.

{Lk}km,

,

PROOF.Let F k 4 {a E A : Lka = O}. Our lemma will be proved when we show that A # up= Fk . First, note that each Fk is a closed linear subspace of A and that Fk # A because L k # 0.We now show that F , is nowhere dense; that is, its closure F k has no interior points. Suppose that b is an interior point of F k . Since F k = F k , this means that there exists an E E R, such that O(b, &) c F k , where O(b, 6) {a E A : IJa- bll < E}. Also, since Lk # 0, there exists a nonzero c E A such that Lkc # 0. Then, d = b + ~ l l ~ l-'cl E O(b, E). Upon applying Lk to d and using the fact that Lk b = 0,we obtain Lkd=3EllCll-1LkC #o. This is a contradiction, and therefore F , is truly nowhere dense. Baire's category theorem (Appendix C13) now implies that A # F , . 0

,

+

u

Theorem 8.5-2. W is a linear translation-invariant causal mapping of 9(H) into [9;HI such that % = 0 ifand only if,for all 4 E 9(H),

8.5.

LOCAL MAPPINGS

159

where Po , PI, .. .,P,, E [ H ; H ] and Pk'

k = 0,1, . ..,n.

= (-l)k"Pk,

(5)

Note. As before, the prime denotes the adjoint operator.

PROOF.If % is defined by (4), then it is clearly a linear translation-invariant causal mapping on 9 ( H ) into 9 ( H ) . Some integrations by parts show that 8 = 0. Conversely, assume that % is a linear translation-invariant causal mapping of 9 ( H ) into [9; HI and such that 8 = 0. For any fixed T E R and a, b E H , consider the mapping

e b ((wa)(T), b),

e E D.

(6)

Since % is a continuous linear mapping of 9 ( H ) into &(H), (6) is a complexvalued distribution. Its support is {T} because % is local. Therefore, by a standard result for complex-valued distributions (Zemanian, 1965, Theorem 3.5-2), ( w e a m b) =

i

k=O

a,(a, b ) e w ) ,

(7)

where the ak(a, b) denote complex numbers and satisfy the following condition. For each fixed choice of the pair a, b E H, all but a finite number of the tlk(a, b) are equal to zero. This finite set will change in general for different choices of a and b. However, the ak(a,b) do not depend on T, because of the translation invariance of %. Moreover, it is not difficult to show that each is continuous and sesquilinear on H x H supplied with the product topology. Therefore, according to Appendix D15,there exists a Pk E [H; HI such that 6 ) = (pk a , b). Thus, (7) becomes

We now show that there exists a finite integer n such that (Pka, b) = 0 for all k > n and all a, b E H. If this is not true, there exists a subsequence {Q,}Y=, of {pk}& such that Q j # 0 for all j . By Lemma 8.5-1, there is an a E H such that Q j a # 0 for all j . But ( Q j a , -)E [H; C], and so, by Lemma 8.5-1 again, there is a b E H such that ( Q j a ,6) # 0 for all j . This contradicts the fact that only a finite number of coefficients in the right-hand side of (8) can be nonzero. It now follows that, for any E 9 0 H ,

+

8.

160

THE ADMITTANCE FORMULISM

Since 9 0 H is dense in 9 ( H ) , (9) remains true for all 4 E 9 ( H ) . This establishes (4) since b and Tare arbitrary. We turn now to the proof of (5). Set Qk Pk + (- l)kpk’.Some integrations by parts show that, for any 4, $ E 9 ( H ) ,

The right-hand side is equal to zero by hypothesis. Thus, for any a, b E H and any 8, A E 9,

11

(Qka, b)O(”(t)yt)dt = 0.

We may apply Parseval’s equation [see (3) of Section 8.41 to this and employ the fact that (gO(”)(q)= (iq)k&q)to get

1

(Qk

4 b)(irl)k’%ll)X(rl)drl = 0

(1 1)

for all 8, 1 E 2. According to Lemma 8.4-2, 2 . 9’ is dense in 9. Therefore, (1 1) continues to hold with %I replaced by any x E 9. Consequently,

2 (&a,

b)(ir])k= 0

k=O

for all q. Whence, ( Q k a ,b) = 0 for all a, b E H , which implies that Qk = 0. This proves (5). 0 Problem 8.5-1. Verify that the quantities c(k occurring in the proof of Theorem 8.5-2 are continuous sesquilinear forms on H x H and do not depend on T. Problem 8.5-2. Let !Jl be a continuous linear translation-invariant local mapping of 9 ( H ) into [ 9 ;HI. Show that

where Q k E [ H ; HI.

8.6. POSITIVE SESQUILINEAR FORMS ON 9 X 9

We shall use the Bochner-Schwartz theorem to obtain a representation for positive, separately continuous, translation-invariant sesquilinear forms on 9 x 9. See Appendices A6 and F1 for the definition of a positive, separ-

8.6.

POSITIVE SESQUILINEAR FORMS ON

9

X

9

161

ately continuous, sesquilinear form on 9 x 9. As for translation invariance, we have the following. Definition 8.6-1. Let X be a space of functions from R into H that is closed under translations (i.e., if 4 EX, then or+E X for every T E R). A sesquilinear form 23 on X x X is said to be translation invariant if 23(0,4,

for all

4, tb,

0 7

ICI)

= 23(4?$1

E X.

Given any y

E

[9;C ] , define the mapping 23 on 9 x 9 by

-

B(e, A) A ( y , e * A T ) ,

e, A E 9,

(1)

where At(t) A( - t). Then, !3 is a separately continuous translation-invariant sesquilinear form on 9 x 9. Indeed, the sesquilinearity is clear. The separate continuity follows from the fact that both O H 0 * It and AH 0 * At are continuous mappings of 9 into 9. Finally, the translation invariance follows from the easily established identity

o * V,

=

* (o,~)f

Conversely, every separately continuous translation-invariant sesquilinear form on 9 x 9 has the representation (1). Indeed, by Theorem 4.5-1, there exists a unique f E [ g R 2 C ;] such that

w 0 ,4= ( f ( t ,

4,e ( t ) W >

for all 8, 1 E 9.From the translation invariance of 23 and the totality in gR2 of the set of functions of the form O ( t ) A a , we see that f ( t , X) = f ( t - 7,X

- T)

for every T E R. We may now proceed exactly as in Section 5.10 to show that

w e ,4= ( Y * A, e>,

(2)

where y E [ 9 ;C ] is uniquely determined by 23. A simple manipulation converts the right-hand side into ( y , 0 * At). Thus, we have established the following.

Theorem 8.6-1. 23 is a separately continuous translation-invariant sesquilinear form on 9 x 9 $and only if 23 has the representation (l), where y E [ 9 ;C ] . y is uniquely determined b y 23, and conversely. If, in addition, 23 is positive, we have the next result.

8.

162

THE ADMITTANCEFORMULISM

Theorem 8.6-2. 23 is a positive, separately continuous, translation-invariant sesquilinearform on 9 x 9 ifand only if23 has the representation

B(0, A) =

J

R

dpt O(t)X(t),

0, A E 9,

(3)

where p is a temperedpositive measure on the Borel subsets of R . PROOF. The “ i f ” part of this theorem follows quite readily. Indeed, the sesquilinearity is again clear. The separate continuity follows from the continuity of the Fourier transformation from 9 into Y and the fact that p is a tempered positive measure. The translation invariance is an immediate result of the identity (@,8)(w) = e-iw‘O(w). Finally, the positivity of 23 follows from the positivity of p because

s(e,0) = j d p t J 8 ( r ) I ’ 2 0. For the “only if” part of this theorem, we first invoke Theorem 8.6-1 to obtain (2). Upon setting 0 = X E 9, we may write

1( y * 8)O dt

=(y

* 2, A)

= B(A,

A) 2 0.

Thus, y E [9; C] is positive-definite. By the Bochner-Schwartz theorem (Theorem 8.4-l), there exists a tempered positive measure p on the Borel subsets of R such that

B(0, A) = ( y , 0 * At) But G(0 * A t )

=

=

J dp g(0 * A’).

03(At) = Oj.This completes the proof.

0

8.7. POSITIVE SESQUILINEAR FORMS O N 9 ( H ) x 9 ( H )

Our purpose in this section is to extend Theorem 8.6-2 to certain sesquilinear forms on 9(H)x 9 ( H ) . 0 will denote the collection of all Borel subsets of R,and 0, will denote the set of all bounded Borel subsets of R. In Definition 2.3-1, we choose 0,as the collection of all Borel subsets of { t : I t I I k } , where k = 1, 2, . . . . Definition 8.7-1. A tempered PO measure P is a a-finite PO measure on

0, such that g&t)

(I

+

lt12q)-’

is integrable with respect to P for some nonnegative integer q.

8.7.

POSITIVE SESQUILINEAR FORMS ON

9(H)x 9(H)

163

As in Section 2.6 [see Equation (10) there], we set

Bgq(H) { f

E

B ( H ) :g;

YE9 6H } .

Lemma 8.7-1. For each nonnegative integer q, Y ( H ) c Bg4(H).Moreouer,

4 H g ; ' 4 is a continuous linear mapping of Y ( H ) into B 0 H . PROOF. Let

4 E Y ( H ) and consider the Fourier inversion formula:

4(t) = (1/2n)

I &w)eiW' d o .

We know that 6 E Y ( H ) c L,(H). By Theorem 2.5-1, 4 E 9 6 H . Also, by (6) of Section 2.5, 1141115 I I $ l l L , , where 11 . is the norm for ' 3 6 H (see Corollary 2.4-la). Hence, 1 1 4 1 1 15

1 II

$(W)llH

R

dw

Since the Fourier transformation is an automorphism on Y ( H ) , this shows that the canonical injection of Y ( H ) into '3 6 H is continuous. Our lemma now follows from the fact that multiplication by g; is a continuous linear mapping of Y ( H ) into Y ( H ) . 0 By virtue of this lemma and Definitions 2.6-2 and 2.6-3, the integrals

have meaning for any 4, t+b

E

Y ( H ) and any tempered PO measure P

Theorem 8.7-1. If P is a tempered PO measure, then

4

-1

dP, 4 ( t )

(1)

is a continuous linear mapping of Y ( H ) into H . Moreover,

(2) is a positive continuous sesquilinear form on Y ( H ) x Y ( H ) . Note. The mapping (1) is called the tempered [ H ; HI-valued distribution generated by P.

8.

164 PROOF.

THE ADMITTANCE FORMULISM

As in Section 2.6, we set

Q(E) 4

I

E

df',

s,(t),

where E is any Bore1 subset of R and q is chosen large enough to ensure that co.We may then write, for any r$ E Y ( H ) , 11 Q(R)IllH;

-=

II Q ~ ~ ~ I I ~ ~ ; ~ ~. I I ~ ~ ~ ~ ~ ~ I ~ ' O

5

[See (2) of Section 2.5.1 This result combined with Lemma 8.7-1 implies the first conclusion. With respect to the second conclusion, we already know that (2) is a positive sesquilinear form on ggs(H)x 9 J H ) (see the paragraph before Definition 2.6-1) and therefore on Y ( H ) x Y ( H ) . To show its continuity, we invoke (2) of Section 2.6 to write, for any 4, cc/ E Y ( H ) ,

I

1I

jd(pfNO7cc/(t)) = /d(Q,[s,(t)l- ''2r$(0, [ g & W '/2$(t>) 5 IIQ(R)II Il(1 + t2q)'/24(t>ll 1 Il(1

I

+ t2')'/2$(t>ll

1.

(3)

We now appeal again to Lemma 8.7-1 to complete the proof. 0 Theorem 8.7-2. With P again being a tempered PO measure, set

$1 4

/ d(P,dXt), W)?91 $ R

EY(H).

(4)

Then, 'illis a positive, continuous, translation-invariant, sesquilinear form on Y ( H ) x Y(H). PROOF. Sesquilinearity is again clear. The translation invariance follows readily from the identity

(80,r$)(t) = e-"'c$(t). For the continuity, combine an estimate like (3) with the facts that &-+g,-'+ is a continuous linear mapping of Y ( H ) into 9 6H and the Fourier transformation is an automorphism on Y ( H ) . The positivity is asserted by Theorem 8.7-1 since $ E Y ( H ) . 0

4,

The principal theorem of this section is the following converse to Theorem 8.7-2 due to Hackenbroch.

8.7.

POSITIVE SESQUILINEAR FORMS O N

9(H)X 9(H)

165

Theorem 8.7-3. Every positive, separately continuous, translation-invariant, sesquilinear form % on 9 ( H ) x 9 ( H ) has the representation

~ 4II/) ,= J’ d(Pr

4, II/ E WH),

$(t>>,

R

(5)

where P is a tempered PO measure. P is uniquely determined by PI.

PROOF. (i) Assuming for the moment that the representation ( 5 ) holds true, we first show that the tempered PO measure P is uniquely determined by ‘u. Let 4 = Ba and II/ = Ab, where 8 , I E 9 and a, b E H . By virtue of (1) of Section 2.6. Definition 2.6-3, and Theorem 2.3-4, we may write

BI(Bu, Ab) = J’d(P, 8(it)~, X(t)b)

By (3) of Section 2.3, this becomes

BI(Ba, Ib) = d(P, a, b)O(f)X(t),

(6)

where 8, E 2’. Now, 2’ * 2’ is dense in 9’ according to Lemma 8.4-2. Moreover, it follows from Theorem 8.7-1 that

c j d(Pra, b ) ~ ( t ) ++

is a continuous linear mapping of Y into C. Hence, given any a, b E H , 2t determines d(P,a, b)c(t) for all C E 9’.This means that (P,a, b) is uniquely determined as a a-finite complex measure on the bounded Bore1 sets in R (Appendix (312). Since this is true for every a, b E H , % uniquely determines P on those sets. A) A a(&, Ab), where 0, A E 9. (ii) Let a, b E H as before. Set Then, BI,, b is a separately continuous translation-invariant sesquilinear form on 9 x 9. Upon setting ’u, = a,,, and using the polarization equation (Appendix A7), we may write %‘u,,b

= &[PI,+, - % a _ ,

+

i%,+ib

- i%,- ib] .

Note that PI, is positive on 9 x 9. So, by Theorem 8.6-2, for each a E H , there exists a tempered positive measure p a such that

Upon setting

166

8.

THE ADMITTANCE FORMULISM

we have that p a , b is a tempered complex measure and

Note that B I uniquely determines Pl,, b for any given a, b E H . Hence, by the argument in part (i) of this proof, p a , b is uniquely determined by the given Iu, a, and 6 . (iii) We now show that, for each a, b E H , po,b has the representation p a , b ( E ) = (P(E)at

b),

(8)

where E is any bounded Bore1 set in R and P ( E ) E [ H ; H I , . First, note Bla, b(O, A) is sesquilinear in its dependence on a that, for any fixed 0, A E 9, and b. It follows from (7) that {a, b}Hp a , b(E) is a positive sesquilinear form on H x H because p a is a tempered positive measure. Furthermore, by Lemma 8.4-3, there exists a 8 E 9such that I g ( t ) I 2 1 for all t E E . So,

Now, '21 is separately continuous on 9 ( H ) x 9 ( H ) and therefore, according to Appendix F2, continuous. Hence, p,(E) IB1(0a, 0a) I Mllall', where M does not depend on a. Therefore, by Appendix D15, (8) holds, and P ( E ) E [ H ; HI,. (iv) Our next objective is to demonstrate that P is tempered. Since p a , b is a a-finite complex measure, it follows from Theorem 2.2-1 that P is a a-finite PO measure on 0,. We have to prove that gq is integrable with respect to P for some q. We start by showing that the mapping

is continuous from Y x H x H into C , as well as linear with respect to O and a and antilinear with respect to b. (It is understood that 9 'x H x H is supplied with the product topology.) Since both Y and H are Frtchet spaces, we need merely establish that the mapping (9) is separately continuous (Appendix F2). Since p a , b is a tempered complex measure, the mapping is continuous and linear on Y for fixed a and b. Now, assume that 0 and b are fixed and define the mapping Fe, b : H C by

-

For the moment, assume in addition that

8.7.

POSITIVE SESQUILINEAR FORMS ON

9 ( H ) x Q(H)

167

Then, by (7)) FO,

1

= dpo, b , I

= %a,

= %(la, lb)*

b(c,

This implies that F 0 . b is continuous and linear because % is separately continuous and sesquilinear. We now invoke Lemma 8.4-2; thus, for arbitrary 8 E 9, we can choose a sequence {6k}p=, such that each Ok has the form (10) and O k + 8 in 9. By what we have already shown, F e k , b E [H; C] and Fek,b(a) + Fe, b(a) for each a E H . It follows that Fg, b E [ H ; c]also (Appendix 011). A similar argument shows that the mapping (9) is continuous and antilinear with respect to b. We can summarize our results so far as follows. There exist two nonnegative integers q and k and a constant M > 0 such that, for all a, b E H with I(al(I 1 and (Ib((5 1 and for all 8 E Y with max sup 1(1 OspSk I e R

we have that

Ij

+ I t12q)e(p)(t)lI 1,

dkz.b, t e ( t )

1

(11)

M-

(See Appendix F2.) Now, let 8 E 9 be nonnegative and such that O(t) ej(t) = ( I

+ p12q)-18(j-1t),

=

1 for I tI I 1. Set

j = 1 , 2 , ...

.

Then, sup{l(l+ I t 1 2 q ) 8 y ) ( t ) : t E R , O I p S k , j = 1 , 2 , ...} P C < C O . So, by (1 l), for all a E H with llall I 1 ,

1

oI where again p a , I

dpo,

p a ,a ,

e j ( t ) IC M ,

j

=

i , 2 , ...,

So,

This proves that, for all j ,

/IS,

tlsj

d P 1 ( 1 + I t l 2 q ) - 1 [~H~; HI

ICM.

(See Appendix 015.) By Theorem 2.3-1, g, is integrable with respect to P. Thus, P is tempered.

8.

168

THE ADMITTANCE FORMULISM

(v) So far, we have shown that

a(&,I b ) = / d ( P , a , b)&t)X(t>

for all 0 , A E 9 and all a, b E H . From the sesquilinearity of 91, (1) of Section 2.6, and Definition 2.6-1, we see that (5) holds for all 4, $ E 9 0H . By the hypothesis and Appendix F2, 2l is continuous on B(H) x 9 ( H ) . Also, by Theorem 8.7-2, the right-hand side of (5) defines a continuous mapping on Y ( H ) x Y ( H ) and therefore on 9 ( H ) x .9(H). But, since 9 0H is dense in 9 ( H ) , 9 0H x 9 0H is dense in 9 ( H ) x 9 ( H ) . Therefore, ( 5 ) holds for all 4, $ E 9(H). 0

8.8. CERTAIN SEMIPASSIVE MAPPINGS OF 9 ( H ) INTO b ( H )

In this section, we study the properties of certain mappings of 9(H)into 6 ( H ) , which occur in subsequent realizability theorems. For any H-valued function 4 on R , we set

4AX) 4 40 - x) and

Lemma 8.8-1. For any

E

9 ( H ) andanypositivep,

m

qP$,(q) = i p

4(p)(t- x)(e-IXq - 1) d x , -m

This lemma can be established by integrating by parts.

Lemma 8.8-2. Let P be a PO measure and set j(x)

Sm -m

dP,,e'Xq.

8.8.

Also.for cjJ

E ~(H),

OF

SEMIPASSIVE MAPPINGS

~(H)

INTO cff(H)

169

set

(9Jl l cjJ)(t)

~ f~J(X)cjJ(t

(9Jl 2 cjJ)(t)

~ foo

- x) dx

(1)

j(x)cjJ(t - x) dx.

(2)

and

o

These equations are equivalent to

= foo

(9Jl I cjJ)(t)

dP,J>t(-,,)

(3)

-00

and

= foo dP~$[tl(-").

(9Jl 2cjJ)(t)

(4)

-00

Moreover,

fOO

(9Jl I cjJ(t), t{J(t)) dt

= foo d(P~$(, ),

-00

$(,,)),

cjJ, t{J

E

~(H),

(5)

-00

and

Re

r

(9Jl 2 cjJ(t), cjJ(t)) dt =

t foo d(P~ $(,,), $(,,)),

-00

cjJ

E ~(H),

r E R, (6)

-00

where in the last equation, t{J(t) ~ {cjJ(t), 0,

t s; r, t > 0.

(7)

PROOF: Equations (3) and (4) follow directly from Theorem 2.5-2, which also asserts that the integrals in (I) and (2) exist as Bochner integrals. It can be shown that x H j(x)cjJ(t - x), roll cjJ, and rol2 cjJ are all continuous H-valued functions. (See Problem 8.8-1.) Consequently, the left-hand sides of (5) and (6) exist. The right-hand sides of (5) and (6) also exist by virtue of Theorem 2.6-1. Moreover,

J

(rollcjJ(t), t{J(t)) dt

=

R

J (J R

=f =

dt

j(x)cjJ(t - x) dx, t{J(t)) dt

R

J(j(t - x)cjJ(x), t{J(t») dx

J J(J dP~ dt

ei(l-x)~cjJ(x),

t{J(t)) dx.

8.

170

THE ADMITTANCE FORMULISM

By Theorem 2.6-1 again, the right-hand side is equal to

1d(P, dxd,

&I>).

This verifies (5). Next, consider

f

Ir(4)

4(t),4(t))d t

(9~12

-m

=

1' f ( A t dt

-m

- x)4(x),

4 0 ) dx.

-m

Since the values of P, are positive operators and therefore self-adjoint, it follows that the adjoint ofjix) isj( -x). By using this fact, we see that

Upon reversing the roles of t and x in the right-hand side, adding I,(+) to -

Zr(4),and

using (7), we get

2 Re I,(+)

=

1'

dt

-m

j' ( A t - x > 4 ( x ) , 4(0) dx -m

An application of Theorem 2.6-1 as before yields (6). 0

In the rest of this section, Q is a tempered PO measure. Also, p where q = I , 3, 5, . . . ; hence, ip = - 1 . Set P(E) =

E

dQ,U

= 2q,

+ vP)-',

+

where E is any Bore1 set in R and p is chosen so large that (1 vP)-' is integrable with respect to Q. Thus, P is a PO measure according to Theorem 2.3-3. We define j as in the preceding lemma. Also, 23 and 9 denote the sesquilinear forms defined in Section 8.3; namely, for any operator ill from g ( H ) into L?'(H),

8.8.

SEMIPASSIVE MAPPINGS OF

Theorem 8.8-1. For any

9 ( H ) INTO € ( H )

171

4 E Q ( H ) , set

J-m

J-m

and

Then, !TIl and W, are linear translation-invariant semipassive mappings of 9 ( H ) into € ( H ) . In addition, !TI2 is causal on 9 ( H ) . Finally, for iN = W, or %=!TI2,

for all

4, $ E 9 ( H ) .

PROOF. As before, the integrals in (8) and (9) exist as Bochner integrals, and x w j ( x ) b ( t - x), W,4, and !TI2 4 are continuous H-valued functions. Clearly, !TI1 and W 2 are linear and translation invariant, and in addition %, is causal. If we can show that W land !TI2 are semipassive on 9 ( H ) , then it will follow from Theorem 8.3-1 that W, and !TI2 are continuous from 9 ( H ) into [ 9 ;HI. This will imply in turn that !TIl and ‘92, are convolution operators (Theorem 5.10-1) and therefore map 9 ( H ) into € ( H ) (Theorem 5.5-1). First, consider the case of W = !TIl. An application of Theorem 2.5-2 yields

(W14)(t)=

eiqX4(t- x ) dx

dP, R

+

R

I I dP,

R

For each k

=

(1 - einx)+(P)( t - x ) dx.

(11)

(1 - eiqx)@p)(t - x ) dx.

(12

R

1 , 2, . . . , set

+ 1- k dP, k

R

By using Theorems 2.2-3 and 2.5-1 and the extension of (2) of Section 2.5 onto 93 6H 3 !3(H), we obtain the estimate

I(%,, k 4(t),$(t)) 1 5 llP(R)ll[H;HI jR [II$(X)IlH + 2 ~ ~ ~ ( p ) ( x ) ~ / H l x dxll $(t)llH.

where

4, $ E 9 ( H ) .

(1 3)

8. THE ADMITTANCE FORMULISM

172

By Theorem 2.5-2 again, we may also write

It is not difficult to show that the quantities in both pairs of brackets on the right-hand side converge in the strong operator topology as k + 03 uniformly k $ ) ( t ) -, (g1$)(f) in H for each for all x E R.We can infer from this that (g1, t E R. This result in conjunction with (13) allows us to invoke Lebesgue's theorem of dominated convergence (Appendix G18) to conclude that, as

k-,

03,

I

I

$(t)) dt

(gl,k$(r),

R

(gl$(r),

R

$(l)) dt.

(15)

On the other hand, by Lemma 8.8-1, Definition 2.6-2, and Theorem 2.5-2 again, k (%I,

k

$)(t) =

-k

k

dP,(l 4- vIp)6t( - 'I)=

-k

dQ, 6t( - ?).

Therefore, by Lemma 8.8-2,

Im(fll,k4(0,NO)dt

=

-m

-k

d(Q,6('1)~$<'I)),

According to Problem 2.6-4, as k -+

1k

k

4, $ E W O .

(16)

03,

d(Q, 6(rl),$<'I>)-,J m d(Q, ~ ( ' I L$<'I)).

(17)

-m

Upon combining (15)-417), we get

$1

I(%19(t), R

$(t)) dr =

jmd(Q,$(v), -m

$<'I>).

This implies (10). Indeed, the values of Q are positive operators and therefore self-adjoint. Hence, in view of Definition 2.6-3 and (1) of Section 2.6, we may write in effect

BB,($, 4) = /(%$, $1 dt = Sd(Q6,6> = 146, Q6) = Sd(Q$,

$1 =

I(%$,

$) dt

=

!%I~($, $1.

But, for 4 = $, the right-hand side of (10) is nonnegative. Hence, Rl is semipassive on 9(H).

8.8.

SEMIPASSIVE MAPPINGS OF

9 ( H ) INTO b ( H )

173

We now take up the case where % = !R2. Again by Theorem 2.5-2, (%2

4)(t) = 2 jRdP,

W

+2

eiVx4(t- x ) d x 0

sR

m

dP, 0

(1 - eiqX)@p) ( t - x ) d x .

Following our previous procedure, we set

+2

1

k

d P , s W ( l - e hX) + ( P ) ( t- X) d x 0

-k

and conclude again that

ask+co. As the next step, we shall show that

We now set 2, i r j ! k dP,( -q)"-'. Since the values of P are positive operators and therefore self-adjoint, it follows that 2,' = (- l)Tr, where r = 1, . . . , p - 1. Upon integrating by parts, we see that Re

/ (Zr

"(t), $(t)) d t = 0.

$J(~-

R

We infer from this result and (20) that

s

-t Re R ('%, k d'(th

cb(t>) d t = Re

/ ( f dQ, R

-k

&tl(

- q), $ ( t ) ) dt.

Since Q is a PO measure on [ -k, k], we may invoke (6) and take than any support point of 4 to equate the right-hand side to

-t

-k

T

larger

d(Q, 6(4,6Cd).

This establishes (19). Observe that (17) still holds in the present case. The combination of (17)-(19) establishes

%d4,4)

Re

R

(% 4(t), 4(t)) dt

=

/

R

d(Q, &v), ~ ( Y I ) ) .

(21)

174

8.

THE ADMITTANCE FORMULISM

Since the right-hand side is nonnegative, we have hereby proved the semipassivity of %2 on 9 ( H ) . Finally, we need merely apply the polarization equation (Appendix A7) to both sides of (21) in order to get (10). 0

’-

Problem 8.8-1. With P being a PO measure, prove that k

d P , eiqx--+

k

1

m

d P , e’“”

-m

in the strong operator topology uniformly for all x E R. Then, prove the assertion made in the proof of Lemma 8.8-2 that x H j ( x ) 4 ( t - x), !lJ331,$, and %N24are all continuous H-valued functions. Also, prove the assertion made in the proof of Theorem 8.8-1 that (%,, k 4)(t)+ (gz,4)(t) in H for every t E R.

8.9. AN EXTENSION OF THE BOCHNER-SCHWARTZ THEOREM

The Bochner-Schwartz theorem (Theorem 8.4-1) gives a representation of any complex-valued positive-definite distribution as the Fourier transform of a distribution generated by a tempered positive measure. Hackenbroch’s extension (Hackenbroch, 1969, Corollary 3.5) of this representation to operator-valued distributions is the subject of the present section. A similar extension is given by Kritt (1968). Definition 8.9-1. An f E [ 9 ( H ) ;HI is said to be positive-dejinite if, for all

4 EW H ) ,

Theorem 8.9-1. Corresponding to each positive-dejinite f E [ 9 ( H ) ;HI, there exists a tempered PO measure M such that,for every 4 E 9 ( H ) ,

Note. The customary definition of the Fourier transform of a complexvalued distribution (see, for example, Zemanian, 1965, p. 203) can be extended to the members of [ 9 ( H ) ;HI. By virtue of this, this theorem can be restated by saying that every positive-definite distribution in [ 9 ( H ) ;H ] is the Fourier transform of a tempered [ H ; HI-valued distribution generated by a tempered PO measure. In this regard, see also Theorem 8.7-1.

8.10.

CERTAIN CAUSAL SEMIPASSIVE MAPPINGS

175

PROOF.Set 3 = f *. Then, 3 is a continuous, linear, translation-invariant mapping of 9 ( H ) into b ( H ) . As usual, set

Then, 23 is a positive, separately continuous, translation-invariant, sesquilinear form on 9 ( H ) x 9 ( H ) . So, by Theorem 8.7-3, there exists a unique tempered PO measure Q such that

W 4 ,$1 =

1 R

d(Q, &I), $),

4, @ E g ( H ) .

(3)

Now, starting with Q , we proceed as in Section 8.8 to define P and j . We then define by (8) of Section 8.8. As was shown in the proof of Theorem 8.8-1, BR,($, $) is equal to the right-hand side of (3). Therefore,

j$W, *) dt = J (WJ, *) d t . R

Upon replacing $ by Oja, where a E H , j = 1, 2, . . . , and O j d a,S for an arbitrarily chosen z E R , we see that '3 = S l .Furthermore, (1 1) of Section 8.8 and Lemma 8.8-1 show that

(%4)(0 = JR dQrl&(-?I. Since 4Jx)

4(f - x) and

$(x)

+(-x),

we may write

( f , 4 >= (fn6)(0) = (%6)(0)

=

J

R

6,= 4 and

dQd%-r~).

This yields (2) when we set M(E) A Q( - E ) for any bounded Bore1 set E . 0 Problem 8.9-2. Show that, i f f (2), then f is positive-definite.

E

[ 9 ( H ) ;HI possesses the representation

8.10. REPRESENTATIONS FOR CERTAIN CAUSAL SEMIPASSIVE MAPPINGS

We now develop Hackenbroch's representations for linear, translationinvariant, causal, semipassive mappings on 9 ( H ) . The first result is a timedomain characterization.

176

8.

THE ADMITTANCE FORMULISM

Theorem 8.10-1. % is a linear, translation-itioariant, causal, semipassioe mapping on 9 ( H ) ifand only if,f o r every 4 E 9 ( H ) ,

+ rm[j(0) - j ( ~ ) ] # ~ ) (-t x ) d x , JO

where the following conditions are satisjied: PkE [ H ; HI and P k ' = (- I ) k " P k . (As before, Pk' is the adjoint of Pk .) Also, p = 2q, where q is an odd positive integer. Finally,

J

j(x) =

dP, e"jX,

(2)

R

where P , is a PO nieasure on the Bore1 subsets of R.

PROOF.We first derive (1) from the stated properties of 91. As in the proof of Theorem 8.9-1, 53% is a positive, separately continuous, translationinvariant, sesquilinear form on 9 ( H ) x 9 ( H ) . By Theorem 8.7-3, there exists a unique tempered PO measure Q such that

%(4, ICI) = 4

/

R

d ( Q , 6 ( ~ )$ (~! I ) ) .

(3)

We define P from Q as in Section 8.8, j by (2), and 912 by (9) of Section 8.8. We also set 934 fS2. Upon comparing (10) of Section 8.8 with (3), we see that 8, = 8 %Theorem . 8.8-1 also states that 91 is a linear, translationinvariant, causal, semipassive mapping on 9 ( H ) . Therefore, so, too, is % - 91.Since !&-, = 8n- Bg1= 0, it follows from Theorem 8.5-2 that

[(a- 9JI)@](f)=

f pk 4(k)(t),

k=O

4 E g(H).

This combined with the definition of 9JI yields (1). That any operator satisfying (1) possesses the stated properties follows from Theorem 8.8-1 and the identity Re

IR

(Pk4(k)(t),4(t))dt

which is a result of the condition pk'

= (-

= 0,

l ) k + ' P k .0

We turn to a frequency-domain characterization for the Laplace transform of the unit-impulse response of the operator %. With j defined by (2), we have that

8.10.

CERTAIN CAUSAL SEMIPASSIVE MAPPINGS

177

Moreover, for any q5 E E ( H ) , xHj(x)q5(x) is a continuous H-valued function according to Problem 8.8-1. As a result, the mapping

4 HJom.i(x)+(-v)dx + J m[i(o>- j ( x ~ p ) ( xdx ) 0

is a member of [9Ll(H);H I . Indeed,

and a similar estimate holds for the second integral. With 'illdefined by (I), we have from Theorem 5.5-2 that 91 = y y E [9Ll(H); H ] is defined on any q5 E L f a , b ( H ) with b < 0 < a by

+ /"[j(0) - j ( ~ ) ] q 5 ( ~ ) (dx. x) 0

*, where

(4)

By continuous extension, this equation is seen to hold for all 4 E gL,(H). Clearly, supp y c [ 0, a). By Lemma 7.2-1, we now have that y E [ Y ( O , 00 ;H ) ; HI. Consequently, the Laplace transform Y of y exists and, for every c E C , and a E H , we have ~ ( i > a [(L'y)(i)]a = (y(x), ae-i")

Theorem 2.5-2 allows us to reverse the order of integration. We then integrate with respect to x and note that a E H is arbitrary, to get

These results combined with Theorem 8.10-1 and the uniqueness property of the Laplace transformation yield the following frequency-domain realizability theorem. Theorem 8.10-2. If 91 is a linear, translation-invariant, causal, semipassive mapping on 9 ( H ) , then 92 is a convolution operator y * where y E [QL,(H);H ] satisfies (4). Moreover, Y ( c ) ( S y ) ( [ )existsfor at least all c E C, andsatisfies (5). Here, Pk , P, , and p satisfy the conditions stated in Tlieoren?8.10-1. Conversely, for every function Y having the representation (9,there exists a

178

8.

THE ADMITTANCE FORMULISM

unique convolution operator 91 = J * suclt that (2jj)(() = Y(4') f o r 4' E C, . Moreover, J E [BL,(H); HI, and y is gicen on any 4 E 9L,(H) by (4). Finally, '9l is a linear, tratislatioti-invariant,causal, seniipassive mapping on 9 ( H ) .

8.11. A REPRESENTATION FOR POSITIVE* TRANSFORMS

If in Theorem 8.10-2 the condition of semipassivity on 9 ( H ) is replaced by the stronger requirement of passivity on 9 ( H ) , then the representation (5) in the preceding section takes on a stricter form. In particular, the following additional conditions occur: n = 1, P , E [H; HI,, and p = 2. The proof of this result, which is given in the next section, is based upon a representation due to Schwindt (1965) for certain [H; HI-valued functions, which we call positive*. Dejnition 8.21-2. A function Y of the complex variable 4' is said to be a positive* niapping of H into H (or simply positive") if Y is an [H; HI-valued analytic function on C , such that R e ( Y ( l ) a ,a ) 2 0 for every 4' E C, and a E H.

When H = C, [C; C] may be identified with C, in which case the following classical result is in force. Theorem 8.11-1. F is a coniplex-valued positive* function for all 4' E C , , F adinits the representation

if and

only if,

Mhre X E R , X 2 0, arid 9l is a j n i t e positive measure on the Bore1 subsets of R . F uniquel), determines X aniong the cornplex numbers and 91 among the complex nieasures. A proof of this is given by Akhiezer and Glazman (1963, Section 59). Actually, in their version, N is a bounded nondecreasing function o n R and the integral is interpreted in the Stieltjes sense. However, this is entirely equivalent to the present statement; see Zaanen, 1967, pp. 33-34, 63-64. Schwindt's representation is a generalization of ( I ) to the case where F is [ H ; HI-valued, as follows.

8.1 1 .

A REPRESENTATION FOR

POSITIVE* TRANSFORMS

Theorem 8.11-2. Y is a positive* inupping of H into H [ E C , , Y can be representedby

179

if and only if, for all

wl?ereP , E [ H ; H I , , Po is a skew-adjoint member of [ H ;HI, and P,, is a PO measure on the Borel subsets of R. PROOF.Assume that Y is defined by (2). Since rl

(1 - irl5M5 - irl)

is a member of 9,it follows from (4) of Section 2.2 and Rudin (1966, p. 201) that the integral on the right-hand side of (2) is weakly analytic and therefore analytic on C , . Therefore, so, too, is Y. Furthermore, for all a E H , we have that (P,a, a) 2 0, ( P ( - ) a ,a ) is a finite positive measure, and (Po a, a) is imaginary. Also,

for all ( E C , . It follows that Re(Y(()a, a ) 2 0 for [ E C , and a E H . So, truly, Y is positive*. Conversely, assume that Y is positive* and set

F&a, 6 ) = ( Y ( i ) a , b),

a, b E H .

(3)

Thus, for each fixed 5 E C , , F , is a continuous sesquilinear form on H x H . Also, for fixed a E H , 5 H Fc(a,a ) is a complex-valued positive* function. So, by Theorem 8.1 1-1,

where, for each fixed a E H , X ( a ) is a real nonnegative number and N,(a) is a finite positive measure. The notation N,,(a) denotes the measure E H [N,,(a)](E),where E is any Borel subset of R. Upon setting Po 4 f [ Y ( l )- Y(l)’],we see immediately that Po is skew-adjoint and that

(Poa, a ) = i Im F,(a, a).

(5)

Now, from (3) and (4) and the fact that ,%‘(a)( is uniquely determined by FS(a,a), we see that X satisfies the following two identities. For all a, b E H and B E C, X(B4 =

IB I ’ x ( 4

(6)

8.

180

THE ADMITTANCE FORMULISM

and

X(a

+ 6) + X ( a - 6) = 2X(a) + 2X(b).

(7)

Similarly, N,(Pa) =

I B I",(a>

(8)

and

N,(a

+ b) + N,(a - b) = 2N,(a) + 2N,(b).

(9)

Next, set x(a, 6)

+

# [ X ( a 6) - ~ ( -a6) + iX(a + ib) - iX(a - ib)].

(10)

Any functional on H that takes on only nonnegative values and satisfies (6) and (7) defines through (10) a positive sesquilinear form x on H x H such that x(a, a) = X ( a ) 2 0.

(1 1)

(See, for example, Kurepa, 1965.) Similarly, we define the complex measure Q,(a, b): E H [Q,(a, b)](E)on the Borel subsets E of R by Q,(u, b)

$[N,,(a+ b) - N;(a - b) + iN,,(a + ib) - iN,(u - ib)]. (12)

Again (see Kurepa, 1965), for every E , {a, b}t+ [Q,(a, b)](E) is a positive sesquilinear form on H x H such that

[P,(a, a>l(E>= [N,(a)l(E) 2 0.

(13)

With these results, we see from (4), (1 I ) , and (13) that Re F,(a,

4 = x ( a , 4 + [Q&,

a)l(R>.

Thus, x(a,a> 5 IF,(a,

41 5

IIY(l>ll llal12.

We may now conclude from Appendix D15 that there exists a unique PI E [ H ; H I , such that (P,a, b) = x(a, b) for all a , b E H . In the same way, we see that, for every Borel set E c R, there exists a P,,(E)E [ H ; H I , such that (P,(E)a, b ) = [Q,(a, b)](E). In fact, Theorem 2.2-1 asserts that P, is a PO measure. Upon combining these results, we see from (4) that, for all a E H ,

where P o , PI, and P, satisfy the conditions stated in the theorem. Upon appealing to the polarization equation and Equation (4) of Section 2.2, we finally obtain (2). 0

8.1 1.

A REPRESENTATION FOR POSITIVE* TRANSFORMS

For any a E H and any

f~

E R, , (2)

181

yields

Both the real and imaginary parts of the quantity within the brackets are bounded on the domain {{a, q } : I < a < co,

-00

< q < 0O}

and tend uniformly to zero as f~ + 00 on any compact subset of the q axis. As a consequence, the integral tends to zero and a-'(Y(a)a, a ) tends to (P,a, a). By the polarization equation, therefore, lim a-'(Y(a)a, b ) = ( f l u ,b) a+ m

(14)

for all a, b E H . We shall make use of this result in a moment. When studying real passive operators in Section 8.13, we will meet a special type of positive* functions, namely, the positive*-real functions. In anticipation of this, we determine the special form that Schwindt's theorem assumes for these functions. Assume throughout the rest of this section that H is the complexification of a real Hilbert space H,. Definition 8.11-2. A function Y is said to be a positive*-real mapping of H into H (or simply positiue*-real) if Y is positive* and, for each real positive number a, the restriction of Y(a) to H , is a member of [H,; H,].

Lemma 8.11-1. Let Q be a skew-adjoint member of [ H ; H I . Then, Q is real ifand only if( Qa, a) = 0for all a E H , .

PROOF.If Q is real, (pa,a) is real for all a E H,. But, (Qa, a) is imaginary because Q is skew-adjoint. Therefore, (pa, a) = 0. Conversely, assume that (pa,a) = 0 for all a E H , and set Q = Q , + iQ, , where Q , and Q2 are real. Then, Q' = Q,' - iQ2'. Since Q is skew-adjoint, Q, = - P I ' and Q 2 = Q2'. By the preceding paragraph, ( Q l a ,a) = 0 for all a E H,, so that (Q2a, a) = 0. Since Q , is self-adjoint,

+ ib), a + ib): a, b E H,, /la + ibl/ = I ) , and, moreover, ( Q2(a+ ib), a + ib) is real. Therefore, ( Q2(a+ ib), a + ib) = ( Q 2 a, a) + b, b) = 0. lIQzll

= sup((Q2(a

(Q2

Hence, Q2 = 0, so that Q is real. 0

8.

182

THE ADMITTANCE FORMULISM

Theorem 8.11-3. Y is a positiiv*-real niappiiig of H into H ifarid onlj. fi tlie representation ( 2 ) possesses tile fidlowitig additional properties: Po and PI are real mappings, and P, = P - , . Note. The measure E , P - , ( E ) Li P,( - E ) .

P - , is defined as follows. For any Bore1 set

PROOF.Assume that P o , P , , and Pll have the stated properties. We may set P, = L, iM,, where the measure L, and M, take their values in [H,; H,]. It follows that L , = L - , and M, = - M - , . For i= c E R , , the imaginary part of the integral in (2) is

+

and this is equal to zero by the oddness and evenness of the integrands. Thus, Y ( c ) is real, so that Y is positive*-real. On the other hand, when Y is positive*-real, (14) shows that PI is real. Moreover, for any a E H,, we have ( Y ( l ) a ,a) = ( P l a ,a)

+ (Pea, a> + j" d(P,a, a). R

The left-hand side as well as the first and last terms on the right-hand side are real numbers. Therefore, so, too, is (Pea, a). But Po is skew-adjoint, which requires that (Pea, a) be imaginary. Hence, (Pea, a ) = 0. By Lemma 8.11-1, Po is real. We can now conclude that G(a)

/R

1 - irp LI(P,,0 , 6) 7 = ([ Y(a) - Pla - Po]a, b ) a - 1q

is a real number for all a, b E H , and all a E R , . It follows from the reflection principle that G(i) = G(5) for all iE C, . We infer from this that a,bEH,,

CEC,.

By the uniqueness assertion of -Theorem 8.1 1-1, ( ( P , - P-,)a, 6) = 0 for all a , b E H,, and therefore P, P -,. 0 2

Problem & I / - I . An [H; HI-valued analytic function Y is said to have a pole at a point i E C if, for some neighborhood R of z, for all iE R\(i}, and for some positive integer 11,

8.12.

POSITIVE* ADMITTANCE TRANSFORMS

183

where F, E [ H ; HI and the series converges in the uniform operator topology. The pole is called simple if n = 1. Assume that Y is positive* and has a pole at a point z = iq of the imaginary axis. Show that the pole is simple and F - , E [ H ;H I , .

8.12. POSITIVE* ADMITTANCE TRANSFORMS

We are at last ready to establish the realizability conditions for a linear translation-invariant passive admittance operator. Consider the distribution y E [ 9 ( H ) ;HI defined by

+ Cm[j(0)- ~ ( X ) I ~ ( ~ ) d( Xx ,) ' 0

where P ,

E

[ H ; H I , , Po is a skew-adjoint member of [ H ; H I , j(x)

1 d P , eiqx, R

and P , is a PO measure on the Bore1 subsets of R. As was shown in Section 8.10, y E [ g L , ( H ) ;H I , and ( I ) continues to hold for all 4 E g L , ( H ) .Moreover, the Laplace transform of y is precisely Schwindt's representation for a positive* mapping of H into H . One of the things we shall show is that (1) characterizes the unit-impulse response J' of a passive convolution operator. Lemma 8.12-1. Assume that y E [gL,(H); HI and that J' * is passive on 9 ( H ) . Choose c, d E R sucli that d < 0 < c. Then, y * is passive on L?c.d(H). PROOF.Choose a, b E R such that d < h < 0 < a < c. As was shown in the proof of Lemma 7.4-2, given any 4 E L?c,d(H), we can find a sequence {+j}y= c 9 ( H ) that converges in L?,,b(H) to 4. Upon setting 11/ = y * 4 and $ j = y * c $ ~we , obtain from Theorem 5.5-2 that t,hj 11/ in B ( H ) . The lemma now follows from the estimate --f

-T

"7

184

8.

THE ADMITTANCE FORMULISM

Theorem 8.12-1. If % is a linear translation-invariant passive operator on 9 ( H ) , then % is a convolution operator y * ,where y E [ g L , ( H ) ;H ] satisfies (1). Moreover, Y !i?y exists and is positive.* Conversely,for every positive* mapping Y of H into H , there exists a unique convolutionoperator% = y *such that ( 2 y ) ( [ )= Y ( [ )forall[ E C, . Moreover, y is represented by (l), so that y E [ g L , ( H ) ;HI and supp y c [0, GO). Also, '9l is a linear translation-invariant passive mapping on 9 ( H ) .

PROOF.Assume that the operator '3 is linear, translation-invariant, and passive on L@(H). By Theorem 8.2-1, % is causal on L@(H). So, by Theorem 8.10-2, '3 = y *, where y E [ Q L I ( H ) ; HI and supp y c [0, GO). By Lemma 7.2-1, y E [ Y ( O , G O ; H ) ; HI so that Y 4 i?y exists and has a strip of definition containing C+ . We will now show that Re(Y([)a,a ) 2 0 for all [ E C , and a E H . To this end, we first invoke Theorem 5.5-2 to write ( f l d ) ( t )= ( Y ( 4 , 40 - 4)

for all 4 E Y C , d ( H ) ,where d < 0

-= c. By Lemma 8.12-1, for any T ER,

Fix upon a [ E C , and choose X > T.Also, choose O E Ep such that O(t) = err on -a < t < X and O(t) = 0 on X + 1 < t < GO. Finally, fix c, d E R such that -Re [ < d < 0 c c. Then, 0 E Y cd,. Upon putting 4 = Oa in (2) and noting that 4 ( t - x ) = aei(' - x ) for - 00 < t < T and for all x in some neighborhood of supp y , we can manipulate (2) into

I

T

Re( Y ( [ ) u ,u )

e2

dt 2 0.

- W

This implies that Re( Y([)a,a) 2 0. Thus, Y is positive*. We now appeal to Schwindt's representation and the uniqueness property of the Laplace transformation to conclude that y satisfies (I). This establishes the first half of our theorem. Next, assume that Y is positive*. By Schwindt's representation, Y = Qy, where y has the representation (1). In view of Theorem 8.10-2, the remaining statements of Theorem 8.12-1 are all clear except for the assertion that 3 4 y * is passive on 9 ( H ) . To show this, let T ER , 4 E . 9 ( H ) , and a E H . Since Po is skew-adjoint, (Poa, a) is imaginary, so that

8.12.

POSITIVE* ADMITTANCE TRANSFORMS

185

so that

Now, set

(%,4)(t)

+/

/m.i(x)q5(t - x) dx 0

m

[ j ( O ) - j ( ~ ) I @ ~ ) (t x> dx.

0

The substitution of the definition ofj(x) and an application of Theorem 2.5-2 converts this into (%14)(t) =

J

dP,&t,(-V)

+ JR dP,[-M#J(t) + q2&1(-v)1.

We are using here the notation defined at the beginning of Section 8.8. For any positive integer k , we define !JIl,k by k (%i,k+>(f) -k

q 2 ) - iQk#’(f),

dPq&t](-q)(l

(5)

where Q k = J!-k dP,q. Note that (&a, a) = J!-k d ( ~ , aa)q, , which is a real number. Hence,

By the argument given in the proof of Theorem 8.8-1 [see, in particular, (18) of Section 8.81,

Re

lT

T

(ml,k4(t>,

-m

4(t))d t

Re J-:%l

6(f>,4([>)d t

(7)

a s k -+ 00. We define the measure M on any Borel subset E of [ - k , k ] by

M(E)

J

E

dP,(1

+ q2>

and set M ( J ) = 0 if J is a Borel set that does not intersect [ - k , k]. Upon identifying M with the PO measure P given in Lemma 8.8-2, we may invoke (6) of Section 8.8 in conjunction with ( 5 ) and (6) to write

8.

186

THE ADMITTANCE FORMULISM

In view of (7),

Upon combining (3), (4), and (8), we see that 9 is passive on 9 ( H ) . 0 Theorem 8.12-1 states the fundamental realizability conditions relating to positive* functions. A similar proof for it, which, however, is not based on Theorem 8.10-2, is given by Zemanian (1970a). As is indicated there, we need merely assume that 9 is linear, translation-invariant, and passive on 9 0H in order to establish the first half of Theorem 8.10-2. It should also be noted that Theorem 8.12-1 possesses an extension to n dimensions (that is, to the case where 9 = g R ,is replaced by BR,,)due to Vladimirov (1969a, b).

8.13. POSITIVE*-REAL ADMITTANCE TRANSFORMS

Here we show how the realizability conditions for an admittance operator are sharpened when that operator is real. Once again, we assume that H is the complexification of a real Hilbert space H , . Theorem 8.13-1. If 9 is a linear translation-invariant passive operator on 9 ( H ) and Y'iTl maps 9 ( H , ) into [ 9 ( R ) ;H , ] , then 9 = y *, where y E [aLI(Hr); H , ] satisjies (1) of Section 8.12 with PI, P o , and P , restricted as in Theorem 8.1 1-3. Moreover, Y g f!y is positive*-real. Conversely, if Y is a positive*-real mapping on H into H , then Y = f!y on C , , where y is a member of [ g L , ( H , ) ;H , ] and satisfies ( I ) of Section 8.12 with P , , P o , and P, restricted as in Theorem 8.1 1-3. Moreover, 9 g y * maps 9 ( H , ) into [ 9 ( R ) ;H , ] and is a linear translation-invariant passive operator on 9 ( H ) .

PROOF.We have already seen that every positive* function is the Laplace transform of a unique y E [ g L , ( H ) ;HI given by ( I ) of Section 8.12. We now invoke Theorems 8.1 1-2 and 8.11-3, which give the necessary and sufficient conditions on P,, P o , and P, in order for Y to be positive*-real. In this case, y E [ g L , ( H r ) ;H , ] because the imaginary part of dP,ei'fxequals zero by virtue of the condition P , = P - , . The second half of our theorem now follows from the second half of Theorem 8.12-1. Under the hypothesis of the first half of the theorem, we can prove that y E [ 9 ( H r ) ; H , ] and Y ( o )E [ H , ; H , ] for every o E R, exactly as in the proof of Theorem 7.6-1. Hence, Y is positive*-real, and, by Theorem 8.12-1 again, 9 = y *, where y has the stated properties. 0

8.14.

PASSIVITY AND SEMIPASSIVITY CONNECTION

187

Problem 8.23-2. Assume that % is a linear translation-invariant passive operator on 9 ( H , ) with range in [ 9 ( R ) ;H,]. Show that '3 has a unique extension as a linear translation-invariant passive operator on 9 ( H ) .

8.14. A CONNECTION BETWEEN PASSIVITY AND SEMIPASSIVITY

Some conditions under which a semipassive convolution operator is passive is given by the next theorem, the scalar version of which was given by Konig and Zemanian (1965). Its proof is based upon the following observations. The function j in the representation ( I ) of Section 8.12 is a strongly continuous [ H ; HI-valued function by virtue of Problem 8.8-1. Therefore, j is strongly measurable, and, by the principle of uniform boundedness, it is bounded in the uniform operator topology on every compact set. Thus, in accordance with Problem 3.3-2, j defines a member of [ 9 ;[ H ; HI] (which we also denote b y j ) by means of the equation ( j , o)a

l j ( x ) a e ( x ) dx,

aEH,

e E 9,.

Now,

converges in the strong operator topology and defines a strongly continuous function of x. Therefore, (1) is also a member of [ 9 ;[ H ; HI]. Through an integration by parts, (j:j(t)l+(i)

) 1:

dt, --e(')(x) a =

j(x)aO(x) d x = ( j , 0 ) a .

By our usual identification between [ 9 ( H ) ; HI and [a;[ H ; HI], j F j ( x ) $(x) dx is the value assigned to any q5 E 9 ( H ) by the generalized derivative of (1). In the same way, [ j ( O ) - j ] l + is strongly continuous and therefore a member of [ 9 ;[ H ; HI]. Thus

Jorn[j(o)- ~ ( X > I ~ ( ~dYx X ) is the value assigned to any q5 E 9 ( H ) by the second generalized derivative of [ j ( O ) - j l l + * However, for p 2 4,

8.

188

THE ADMITTANCEFORMULISM

is not the value assigned to 4 by the second generalized derivative of a strongly continuous [ H ; HI-valued function. This is because f 4 D’{[j(O) - jJl+} and therefore D”-’{[j(O) - j J l + }are not strongly continuous. To show this, let m

ek(x)

e - k 2 x 2 / / - m e - k 2 xd2x ,

k

= 1,

2, . . . .

Since Ok E 9’c QL, and [ j ( O ) - j ] l + E [QL,(H);H I , we may write, for any aEH,

((fa,

ek>,

a) = ( ( W [ j ( O ) - j I l

+I, e k > a ,

a ) = ( ( [ j ( o )- j N + , e:’))a, a )

An application of Theorem 2.5-2 and two integrations by parts convert the right-hand side into

1

m

d(P, a, a)q2

Ok(x)eiqxd x .

0

This is equal to

41 d(P,,a, a)q2[exp(- q’/4k2)] erfc( - iq/2k), where erfc denotes the complementary error function (see Zemanian, 1965, pp. 350, 357). But Re erfc( - iq/2k) = 1, and thus (2) clearly does not tend to zero as k + 00. It would have to do so if f A D’{[j(O) - j ] l + } were strongly continuous at the origin because Ok -,6 and supp f c [ 0, 03). Finally, we note again that Y 4& !i?y is positive* if and only if y is represented by ( 1 ) of Section 8.12. In view of these results, we need merely compare Theorems 8.10-1 and 8.12-1 in order to conclude the following. Theorem 8.14-1. Let y E [Q(N); HI. Then, y if y satisfies the following conditions.

+

* is passive on Q ( H ) ifand only

(i) y = PI6(” w ,where PI E [ H ; HI+ , is as usual theJirst generalized derivative of the delta functionaI, w is the second generalized derivative of an [ H ; HI-valued function on R that is continuous with respect to the strong operator topology, and supp w c [0, 03). (ii) y * is semipassive on Q(H).

8.15

189

ADMITTANCEAND SCATTERING FORMULISMS

8.15. A CONNECTION BETWEEN THE ADMITTANCE AND SCATTERING FORMULISMS

With %: U H U and !XI: q H r denoting respectively the admittance and scattering operators for a Hilbert port, the variables u, u, q, r E [ 9 ;HI satisfy v=q+r,

u=q-r.

(1)

If a particular Hilbert port has an admittance operator, need it have a scattering operator as well? No. For, a substitution of (1) into %v = u yields %r

+ r = %q - q.

Upon setting % = - 6 *, we see that only q = 0 will satisfy this relation, whereas any r will do. This means that 2B:q H r does not exist as an operator. A similar manipulation shows that % will not exist as an operator when '123 = -6 *. However, when % is a linear translation-invariant passive operator on Q ( H ) , !XI exists and is a linear translation-invariant scatter-passive operator on Q ( H ) . The converse is not true in general, but it will be true if we assume in addition that I + S(() possesses an inverse in [ H ; HI for each ( E C , , where I is the identity operator on H and S is the scattering transform. To show these results, we first establish a connection between S and the admittance transform Y.

Lemma 8.15-1. Let Y be positive*. Then, for every ( E C , , I possesses an inverse in [ H i HI. Moreover, S ( . ) P [ I +Y(.)]-"I-

is bounded *.

+ Y(() (2)

Y(.)]

PROOF.Fix ( E C, and set F A I + Y ( ( ) . We wish to show that F-' exists. Set b = Fa, where a E H . Then, 2(a,a) I 2 ( a , 4

+ ( Y o , 4 + (0, Y a ) = (6, a) + ( a , b) 5 21(a, 611 S211all llbll,

This implies that F is injective. F ( H ) is a closed linear subspace of H . Indeed, let b be a limit point of F ( H ) and choose a sequence {b,} c F(H) that converges to 6. Then, b, = Fa,, where a, E H . By (3), llan - a m l l

IIF(an - am)II

=

llbn - bmll

-+

0.

8.

190

T H E ADMITTANCE FORMULISM

So, {a,} is a Cauchy sequence and converges therefore to an a E H. Thus, Fa, + Fa by the continuity of F, whereas Fa,, = b, -+ b. Hence, b = Fa E F(H). If F ( H ) were a proper subspace of H, there would be an a E H with a # 0 such that (Fa, a) = 0 (Berberian, 1961, p. 71). That is, ( [ I + Y(c)]a, a) = 0. But this cannot be since Re( Y([)a, a ) 2 0. Hence, F is surjective. Thus, F - ' exists on H. Moreover, F - ' E [H; HI according to Appendix D14. In the same way as in the scalar case, it can be shown that [ I + Y( . )I-' is analytic at every point [ where I + Y([) has an inverse and Y is analytic. (In this regard, see Problem 8.15-1.) Thus, [ I + Y( . )I-' and therefore S are [H; HI-valued analytic functions on C+ . Upon premultiplying ( 2 ) by I + Y and rearranging the result, we obtain Y(Z + S ) = I - S. Thus, for any a E H, Ilal12 - IIS(0all2 = Re([[ - S(i)la, [ I + S(0la) =

Re( Y(O[l

+ S(C)Ia, [ I + S(0la) 2 0.

This implies that IlS([)II I1 for all ( E C+ . We have hereby shown that S is bounded*. 0 Now, assume that '!Jl is a linear translation-invariant passive operator on 9(H). Therefore, 91 = y *, where Y A 2 y is positive*. Moreover, for any t, E 9 ( H ) , u A y * u is a Laplace-transformable member of $B+(H) whose strip of definition contains C, . Consequently, q = t(v + u) and r = +(u - u ) have the same properties. Upon substituting ( I ) into u = y * u, taking Laplace transforms, and rearranging the result, we obtain [I

+ Y(OIN0 = [I - Y(OlQ(5),

iE C+ .

(4)

In view of Lemma 8.15-1, this may be rewritten as

44')= [ I + Y(Ol-l [I -

Y(i)lQ(O

= S(4')Q(09

iE C+ ,

where S is bounded*. Applying the inverse Laplace transformation, we get

r=s*q,

(5)

where s A S - ' S and r and q correspond to the given u E $B(H) as above. We use (5) to define the scattering operator '2u s * on other q and in particular on 9 ( H ) . Since S is bounded*, the next theorem follows immediately from Theorem 7.5-1. Theorem 8.15-1. LcJt 91 br a linear translation-iiii~ariarit passioe operator on Q(H) and let Y be the corresponding admittance transform. Define S by ( 2 ) andset s Q - ' S . Then, XI A s * is a linear translation-iiioariant scatter-passive

operator on 9(H).

8.15.

191

ADMITTANCE AND SCATTERING FORMULISMS

Let us now go into the opposite direction, starting with a given scattering operator 2B and deriving from it an admittance operator '3. We first note that this may not always be possible even when 1113 is linear, translation-invariant, and scatter-passive on 9 ( H ) . A counterexample is 1113 = -6 *. Lemma 8.15-2. Assume that S is bounded* and that, for every [ E C,,

I

+ S([)possesses an inverse in [ H ; HI. Then, Y( . )

[I

+ S( . )]-"I

- S( * )]

(6)

is positive* and satisjies (2). PROOF. We again have that [ I + S( )]-I is an [ H ; HI-valued analytic function on C , . Therefore, so, too, is Y. Since I + S commutes with I - S, ( 6 ) yields

Y(Z + S ) = I - s.

(7)

Now, let b E H be arbitrary and fix [ E C , . Since [ I an a E H such that [ I + S([)]a = b. Therefore,

+ S(4')I-l

exists, there is

Re( W b , b ) = Re( Y[Z + S([>la, [ I + S(0la) = ReW - m > l a ?[ I + S([)Ia> = llallZ - IIS(5)al12 2 0. Thus, Y is positive*. Lemma 8.15-1 now shows that [ I This allows us t o solve (7) to get (2). 0

+ Y([)]-'

exists.

To obtain an admittance operator from a given scattering operator, we proceed in just about the same way as for Theorem 8.15-1 except that now we rely on Lemma 8.15-2. Theorem 8.15-2. Let B ' 3 be a linear transfation-invariant scatter-passive operator on 9 ( H ) , and let S be the corresponding scattering transform. Assume that I + S ( ( ) possesses an inverse in [ H ; HI for every [ E C , . Also, define Y by ( 6 ) and set y 2-l Y. Then, '3 4 y * is a linear translation-invariant passive operator on 9 ( H ) , and the scattering operator generated by '3 in accordance with Theorem 8.15-1 coincides with 2B. Problem 8.15-1. Let G be an [ H ; HI-valued analytic function on C , such that G([)-' exists for all [ E C, . Choose any z E C , . Show that, for all [ E C , such that

II [G(4

- G(r)lG(z)-

II <

1 7

8.

192

THE ADMITTANCE FORMULISM

we have OD

G(()-' - G(z)-'

= G(z)-'

1{[G(z) - G([)]G(z)-'Y. n= 1

Then, using this relation, show that G( )-' is a continuous function on C , . Finally, show that G( * ) - ' is analytic on C , .

8.16. THE ADMITTANCE TRANSFORM OF A LOSSLESS HILBERT PORT

We saw in Section 7.7 how the losslessness of a scattering operator is characterized by the fact that the scattering transform has isometric boundary values almost everywhere on the imaginary axis. A similar connection exists between the losslessness of the scattering operator and the boundary values of the admittance operator. Now, however, the connection is not as complete, since the existence of the boundary values is taken as an assumption. The precise result is stated by the next theorem, (D'Amato, 1971).

Theorem 8.16-1. Let H be a separable complex Hilbert space. Assume that S is a bounded* mapping of H into H and that I + S(c) has an inverse in [ H ; HI forall~EC+.Sets~!i!-'Sand Y ( 0 A [I

+W)l-"I

- S(01,

r

E

c, .

(1)

Also, assume that, for almost all w E R and as r~ -,0 + , Y ( ~ + J i o ) converges in the strong operator topology to Y(iw). I f Y(iw) is skew-adjoint for almost all w, then S(iw) is unitary for the same values of w , and s * is lossless on 9(H). Conversely, gs * is lossless on 9 ( H ) , then Y(iw)is skew-adjoint for almost all w . PROOF.We may rewrite (1) as

+ Y ( 0 l = I - Y ( 0 , r E c, (2) As was indicated in Section 7.7, as r~ -,0 + , S(u + iw) converges in the strong S("

*

operator topology to S(io) for almost all w. It follows from our hypothesis on Y ( l ) and the principle of uniform boundedness that S(()[Z + Y ( [ ) ] converges in the same way to S(iw)[Z + Y(iw)].Therefore, (2) holds almost everywhere when [ is replaced by iw. By Lemma 8.15-2, Y is positive*. This implies that Re(Y(iw)a,a) 2 0 for all a E H and almost all w . By the proof of Lemma 8.15-1, [I + Y(iw)]-' exists almost everywhere. We can now conclude from (2) that

+

S(iw) = [I - ~ ( i w ) ] [ Z ~ ( i w ) ] - '

(3)

8.16.

193

LOSSLESS HILBERT PORT

almost everywhere. After premultiplying and postmultiplying this by I + Y(io),we get

+

[ I + Y ( i w ) ] S ( i w ) [ ~~ ( i w )=] I - [ ~ ( i w ) ] ~ = [ I - Y(io)][Z Y(iw)].

+

Hence,

~ ( i w=) [ I + Y ( ~ w ) ] - ’ [-I Y(iw)]

(4)

almost everywhere. We now employ the identity (F’)-’ = (F-I)‘ for any F E [ H ; H I , where as always the prime denotes the adjoint operator. Setting S’(iw) 4 [S(iw)]’, we get from (4)

+ Y ( ~ ) ] - ’ [-I Y(iw)]}’ = [I - Y(iw)]’{[I+ Y(iw)]’}-’

S’(iw)= { [ I =

[I+ Y ( ~ w ) ][I~(iw)]-’.

The last equality is due to the hypothesis that Y(iw) is skew-adjoint. Combining this result with (3), we get S’(iw)S(iw)= S(iw)S’(iw)= I

almost everywhere. This is precisely the condition that must be satisfied for S(iw) to be unitary (Berberian, 1961, p. 145). Theorem 7.7-1 now shows that s * is lossless on 9 ( H ) . Conversely, assume that s * is lossless on 9 ( H ) . By Theorem 7.7-2, S(iw) is isometric for almost all w , so that S’(iw)S(iw)= 1. By (3) and the identity (F’)- = (F - I)’,

’

[I+ ~ ‘ ( i w ) ] -’ [ Y’(iw)][I ~ - ~ ( i w ) ] [ fY(iw)]-’ + =I. This can be rearranged into

I - Y’(iw)- Y(iw) + Y’(iw)Y(iw)= I

+ Y’(iw)+ Y(iw) + Y‘(iw)Y(iw),

which is the same as Y(iw) = - Y’(iw). In other words, Y(iw) is skewadjoint for almost all w . 0

Appendix A

Linear Spaces

Note. This and the following appendixes survey those standard definitions and results concerning topological linear spaces and the Bochner integral that are used in this book. No proofs are presented since all of them can be found in a variety of readily available books, such as those by Dunford and Schwartz (1966), Hille and Phillips (1957), Horvath (1966), Robertson and Robertson (1964), Rudin (1966), Schaefer (1966), Treves (1967), and Zaanen (1967). Much of the notation used here is explained in Section 1.2, and hence the reader may wish to look through that section before reading these appendixes. The linear spaces occurring in this book are in almost all cases complex linear spaces. The only exceptions occur on the few occasions when we use real linear spaces. For this reason, we will employ the phrase “linear space” to mean a complex linear space, whose definition is as follows.

A l . A collection Y of elements 4, $, 0, . . . is called a (complex) linear space if the following three axioms are satisfied. 194

195

LINEAR SPACES

1. There is an operation +, mapping Y x Y into Y and called addition, by which any pair {4, $} E Y x Y can be combined to yield an element 4 + $ E Y . In addition, the following rules hold: la. 4 + $ = $ + 4 (commutativity). 1b. (4 + $) + 0 = 4 ($ + 0) (associativity). lc. There is a unique element @ E Y such that

+

4 + @ = 4 for every

4 E Y .

Id. For each

4 E Y , there

exists a unique element

-4

E

Y such that

4 + (-4) =@* 2. There is an operation mapping C x Y into Y and called multiplication by a complex number, by which any u E C and 4 E Y can be combined to yield E Y .Moreover, the following rules hold for all a, P E C: an element 2a. 2b.

.(P4)

= (UP)+.

14 = 4 (1

denotes the number one).

3. The following distributive laws hold :

+ $) = a 4 + a$. + P)+ = + P4.

3a. u(+ 3b. (U

A2. The element - $ is called the negative of $. The subtraction of $ from 4 + (- $). Also, @ is called the zero element or the origin of Y ;we usually denote @ by 0. The following rules are consequences of the definition A1 :

4 is defined as 4 - $

(i) (ii) (iii) (iv) (v) (vi)

4 + $ = 4 + 8 implies that $ = 0.

a@ = 0.

04 = @ (here, 0 denotes the number zero). (- 1)4 = - 4 . If a 4 = P4 and = a$ and If

4 c1

# 0, then u = P. # 0, then 4 = $.

A3. A subset % of a linear space “Y is said to be a linear subspace (or simply a subspace) of Y if, for every 4, $ E Y and U E C, we have that 4 + $ E and E %. In this case, it follows that % is also a linear space under Y’srules for addition and multiplication by a complex number. Also, the intersection of any collection of linear subspaces of Y is a linear subspace of Y . A linear combination of elements in Y is a sum q,&, where c(k E C, 4kE “Y, and the summation is over a finite number of terms. The span ofany given subset 0 c “Y is the set of all linear combinations of elements in R. Any such span is a linear subspace of Y . A subspace @ of Y is said to be finite-dimensional if % is the span of a finite number of members of@.

196

APPENDIX A

A finite set { e j } ~ , c , Y is called linearly independent if the equation = a j , pj E C , implies that a j = pi for every j. Every finite-dimensional subspace % contains a linearly independent finite set { e j } whose span is 4'1.

c u j e j c p j e j , where

+

A4. A set R in a complex linear space Y is called conuex if 14 (1 - A)$ E R whenever 4, $ E R and 1 E R is such that 0 2 1 2 1. R is called balanced if a4 E R whenever 4 E R and u E C i s such that JuI I 1. If R is both balanced and convex, it is also called absolutely convex; this occurs if and only if a4 +/?$E R whenver 4, $ E R and a, p E C are such that la1 I 1. Let R be any nonvoid set in the complex linear space Y .The conuex hull of R is the set of all sums 1, &, where 4 k E R, the Ak E R are such that Ak > 0 and 1 Ak = 1, and the summation is over a finite number of terms. A convex set coincides with its convex hull. The balanced convex hull of R is the set of ail sums ak 4 k , where $kE R, the akE C are such that I ak1 I I , and the summation is over a finite number of terms. The set R is said to be absorbent if, for any given E Y , there exists a A E R with A > 0 such that 4 E aR for all a E C with I a I 2 A. The intersection of any finite collection of balanced convex absorbent sets is also balanced, convex, and absorbent.

+

c

1

AS. Let Y and W be linear spaces. A mapping f of Y into W is called linear if, for every 4, $ E Y and a, p E C, we always have that f ( a 4 + P$) = af(4) + Bf($)The set of all such mappings is denoted by L ( Y ;W ) .For anyf, g E L ( Y ; W ) and any a E C, we definef g and af as follows. For any 4 E Y ,( f g ) ( 4 ) A f ( 4 ) g ( 4 ) and (af)(4)4 af(4). As a consequence, L ( Y ;W ) is a linear space. When W is the complex plane C , L ( Y ; C ) is called the algebraic dual of Y and is also denoted by Y * . Each member of L ( Y ; C) is called a linear form on Y . Iff is a linear bijection of Y onto W , its inversef-', which by definition exists, is also linear.

+

+

+

A6. Let %, Y , and W be complex linear spaces. A mapping f of @ x Y into W is called bilinear if 4 Hf(4, $) is linear on 42 for each fixed E Y and $ Hf(4, $) is linear on Y for each fixed 4 E %. B(%, Y ;W )denotes the set of all bilinear mappings of % x Y into W . It becomes a linear space when, for any f, g E B(%, Y ;W ) and any a E C , we define .f g and af by ( f + d ( 4 ,$) = f (4, $1 + s(4,$1 and (af )(4?$) = xf (4, $1 for all 4 E % and $ E Y . We set B(%, Y )A B(%, Y ;C). The members of B(%, Y ) are called bilinear forms on % x Y .

+

197

LINEAR SPACES

A mapping h of Y into Y f is called antilinear if, for any $, 6 E Y and a, P E C , we have that

h(4

+ PO) = Ch($) + Jlh(0).

A mapping f of 49 x Y into W is called sesquilinear if 4 Hf (4, $) is linear on 07L for each fixed $ E Y and $ ~ f ( 4I)) , is antilinear on Y for each fixed 4 E 02.f is called a sesquilinear form on 9 x Y if W = C. ,f is called a positive sesquilinear,form on Y x Y if.f($, 4) 2 0 for all 4 E Y ;in this case, we have the Schwurz inequality:

I f(4,$) I G(4,4)f($?$) for all

4, $ E Y .

A7. For any sesquilinear mapping,f of Y x Y into W , where Y and W are complex linear spaces, we have the polarization identity:

. f ( h $1 = t [ f ( 4 +

$9

4 + 9)-f(4

- $3

4 - $)

+ if(4 + i$, 4 + i$) - i f ( $ - i$, 4 - $)I.

AS. When the complex plane C is replaced by the real line R in axioms 2 and 3 of Appendix A l , the definition of a real linear space is obtained. Except for the definitions of antilinear and sesquilinear mappings and the polarization identity, the preceding discussion carries over to real linear spaces.

Appendix B

Topological Spaces

B1. A topological space is a set Y for which a collection 0 of subsets of V is specified and has the following properties. (i) V and the empty set are members of 0. (ii) Every union of members of 0 is a member of 0. (iii) The intersection of any finite number of members of 0 is a member of 0. The members of 0 are called open sets, and 0 is said to be a topology on V .The complement of any open set in Y is called a closed set. It follows that any intersection of closed sets is closed, and so, too, is the union of any finite number of closed sets. Also, Y and the empty set are closed. Given any set R in V , the largest open set contained in R is the interior 0 of R, and the points of 0 are called the interior points of R. The smallest closed set containing R is the closure of R. Let CR denote the complement of R. Then, B n CR is called the boundary of R. If A is another set in Y and if ii 2 R, then A is said to be dense in R. Y is called separable or ofcountable type if it contains a countable dense subset. On the other hand, Y is called separated 198

TOPOLOGICAL SPACES

199

or Hausdorfand 0 is said to separate Y if, for every pair of points 4, $ E Y , 4 # $, there exist open sets @ and Y such that 4 E @, $ E Y, and @ n Y is empty. B2. A subset R of Y is called a neighborhood of a point 4 E Y (or a neighborhood of a subset -'€' c Y )if there exists an open set A such that 4 E A c R (respectively Y c A c 0). Let N +denote the collection of all neighborhoods of a fixed point 4 E Y .Then, N 4 has the following properties. (i) 4 E R for all R E A'+. (ii) I f Y I A I R n . N + , t h e n A E . N + . (iii) If R, A E .N4,then R n A E N 4 . (iv) If R E .N+, then there exists a A E .N+such that R E .N$for all $€A. The following is a fact: Given any space Y and, for each 4 E Y , given a nonempty collection .Nsof subsets of Y , if conditions (i)-(iv) are satisfied by every N 4 ,then there exists a unique topology in Y that makes .N+the collection of neighborhoods of 4 for every 4 E Y .Because of this, the collection of all neighborhoods of all points of Y can be used as the definition of the topology on Y . A subset a+of the collection N+of neighborhoods of 4 E Y said to be a base of neighborhoods of 4 if, given any R E .N+, there exists a A E &?+ such that A c R. A specification of 9?+for every 4 E Y uniquely determines the topology on Y . B3. Given two topologies 0, and O2 on Y , 0, is said to be stronger or finer than 0, if 0, I 0,.In this case, 0, is said to be weaker or coarser than 0,. Also, the special case 0, = 0, is allowed here. For each 4 E Y , let &?+' and gB,'be bases of neighborhoods of 4 for 0, and 0,, respectively. Then, 0, is stronger than 0, if and only if, given any A E g+',there exists an 52 E g4'such that R c A. B4. Let Y and W be two topological spaces and let f be a mapping of Y into W .f is said to be continuous at 4 E Y if, for any neighborhood A of f(4) E W , there exists a neighborhood R of 4 E Y such that f(R) c A. f is called continuous if it is continuous at every point of Y .The following three conditions are equivalent. (i) f is continuous. (ii) f - '(A) is open in Y for every open set A in W . (iii) f -'(A) is closed in Y for every closed set A in W .

B5. A sequence {&}?=, in a topological space Y is said to conuerge in Y to a limit 4 E Y if, given any neighborhood R of 4, there exists an integer

200

APPENDIX B

K such that 4 k E R for all k > K . In this case, we write 4 k --* 4 or lim 4,' = 4. A mapping f from Y into another topological space W is called sequentially continuous if, for every convergent sequence { 4 k } with limit 4 E Y , we have that f ( 4 k ) -f(4)in W . The continuity off implies its sequential continuity, but the converse is not true in general. If @ is a subset of Y ,"% is said to be sequentially dense in Y if, for every (b E Y ,there exists a sequence {&} c "% such that 4 k + 4 in Y .The sequential density of 92 in Y implies the density of "% in Y , but again the converse is not true in general. B6. Let W be a topological space and Y a subset of W . The induced topology U ion Y is the collection of all sets of the form Y n A, where A is any open set in W . Y with the induced topology is separated whenever W is. The canonical injection of Y into W is the mapping that assigns to each 4 E Y the element 4 E W . If Y has its own topology, say 9, then U is stronger than U iif and only if the canonical injection of Y into W is continuous.

B7. Let Yl, . . . , Y . be a finite collection of topological spaces. The Cartesian product Y 4 Y x . * * x Y,,of these spaces is the set of all ordered .. ., 4,,}, where 4 k E Y k for each k = 1 , . .., n. Now, let n-tuples (b 4 Pk denote the mapping 4- & . Also, for any given 4 E Y , let L?#+ be the collection of all subsets R c Y for which Pk(R) is a neighborhood of 4 k for every k . L?#+ is a base of neighborhdods for a unique topology on Y ,called the product topology. When Y is assigned this topology, it is called the topological product of the Y k . In this case, Y is separated whenever each Y k is. B8. A metric p on an arbitrary set Y is a mapping from Y x Y into the real line R such that the following three rules are satisfied whatever be the elements 4, $, 8, E Y : (i) p(4, $) 2 0; also, p ( 4 , $) = 0 if and only if 4 = $. (ii) P ( 4 , $1 = P($, 4). (iii) $1 I P ( 4 , 0) + P(0, $). A metric space Y is a set with a metric defined upon it. Given any p E Y and a real number r > 0, the open sphere O ( 4 , r ) centered at 4 and of radius r is the set {$ E Y :p ( 4 , $) < r } . We specify a topology on Y by defining each neighborhood of any 4 E F as a set containing an open sphere centered at 4. A topological space is said to be metrizable if its topology can be obtained in this way from a metric. Every metrizable topological space is separated and has a countable base of neighborhoods of each of its points 4, namely {0(4, lln)},",1. A mapping f from a metric space Y into a topological space W is continuous if and only if it is sequentially continuous. Also, a subset "% of a metric space Y is dense in Y if and only if it is sequentially dense in Y . d

4

9

Appendix C

Topological Linear Spaces

Note. In this and the subsequent appendices, we will continue to fix our attention on complex linear spaces. Nevertheless, all the results listed here become valid for real linear spaces upon making the obvious alterations. See Appendix A7.

Cl. By a topological linear space, we mean a linear space Y having a topology such that the algebraic operations of addition and multiplication by a complex number are continuous. That is, the mapping {4, $} H 4 + $ is continuous from Y x Y into Y , and the mapping {a, ~ } H C I I is $ continuous from C x Y into Y when Y x Y and C x Y carry the product topologies. If B? is a collection of subsets of ' 9 and 4 E Y ,B? + 4 denotes the collection of sets obtained by adding 4 to every element in each set of B?. In a topological linear space, B? is a base of neighborhoods of 0 (0 denotes the origin) if and only if B? 4 is a base of neighborhoods of 4. Thus, to determine the topology of Y , we need merely specify a base of neighborhoods of 0.

+

20 1

202

APPENDIX C

C2. Let 8 be a base of neighborhoods of 0 in a topological linear space Y . The following three assertions are equivalent. (i) Y is separated. (ii) R = (0). RE9

(iii) ( 0 ) is a closed set in Y . C3. A locally convex space is a topological linear space having a base of convex neighborhoods of 0. Its topology is also called locally convex. Such spaces can be characterized as follows. If V is a locally convex space, it has a base a of neighborhoods of 0 with the following properties.

(i) If R, A ~ 9 lthen , there exists a E E B such that E c R n A. (ii) If R E 98 and ci E C, where ci # 0, then ciR E B. (iii) Every R E B is balanced, convex, and absorbent. Conversely, if Y is a linear space and a is a nonempty collection of subsets of Y having these three properties, then there exists a unique topology that renders Y into a locally convex space with 9l as a base of neighborhoods of 0. C4. Let Y be a linear space and d any set of balanced convex absorbent subsets of V . Let Y have the topology 0 generated by the base of neighborhoods of 0 consisting of all sets of the form &(), A,, where E E R , E > 0, and 0, Ak is the intersection of a finite collection of A, E d.Then, Y is a locally convex space. Moreover, 0 is the weakest topology under which the algebraic operations of addition and multiplication by a complex number are continuous and every member of d is a neighborhood of 0. Conversely, the topology 0 of any locally convex space V can be produced in this way from a collection d of balanced convex absorbent subsets of Y . C5. Let Y be a complex linear space. A seminorm y on Y is a mapping of Y into R such that, for every 4, $ E Y and every ci E C, we have that y(rw4)= 1 % I y(4) and y ( 4 $) 5 y(4) y($). It follows that y ( 0 ) = 0, y(4) 2 0, and I y ( 4 ) - ?($)I Iy(4 - $). If, in addition, y ( 4 ) = 0 implies that 4 = 0, y is called a norm. For two seminorms y and p on Y , we write y 5 p to mean ~ ( 4 p)( s 4 ) for all 4 E Y . If Y is a locally convex space, a seminorm on Y is a continuous mapping if and only if it is continuous at the origin.

+

+

For any finite collection {y,} of seminorms on V , max, yk is defined by (max, y&4) max, ~ ~ ( and 4 ) is also a seminorm on Y ; moreover, it is continuous whenever each y k is continuous. The same is true for sums x k Y k of seminorms.

TOPOLOGICAL LINEAR SPACES

203

C6. Let Y be a linear space and r any collection of seminorms on Y . Let Y have a topology 0 generated by the base of neighborhoods of 0 consisting of all sets of the form

where E E R , E > 0, and the Yk comprise a finite set of seminorms in r. Then, Y is a locally convex space. Moreover, 0 is the weakest topology under which the algebraic operations of addition and multiplication by a complex number are continuous and every seminorm in r is a continuous mapping. Conversely, the topology U of any locally convex space Y can be produced in this way from a collection r of seminorms on Y . We say that 0 is generated by r and call r a generatingfamily of seminorms for the topology of Y or simply a generating.family of seminorms. If Y is separated, I- is called a multinormfor Y .Actually, Y will be separated when and only when, for every C#J E Y with # 0, there exists some y E r such that Y(4) > 0. converges in the locally convex space Y to a limit 4 A sequence {~$~}p=, if and only if y(& - 4) --t 0 as k + co for every y in any given generating family of seminorms. A base of continuous seminorms for Y is any collection P of seminorms p on Y obtained from any generating family r of seminorms for 0 by setting p = maxk y k , where {yk} traverses all finite subsets of r. P is also a generating family of seminorms for 0. Moreover, P has the property that, given any continuous seminorm '1 on Y ,there exist a constant M > 0 and a p E P such that I M p . C7. Let Y be a locally convex space. The following three assertions are equivalent. (i) Y is metrizable (see Appendix B8). (ii) Y is separated and has a countable base of neighborhoods of 0. (iii) The topology of Y is generated by a countable multinorm (i.e., Y is separated, and its topology is produced from a countable generating family of seminorms).

C8. A subset i2 of a locally convex space Y is called bounded if sup+ y(C#J) < co for every seminorm y in a generating family of seminorms. i2 is a bounded set in Y if and only if, given any balanced convex neighborhood A of 0 in Y , there exists a p E R such that R c p A . The union of any convergent sequence and its limit is a bounded set. C9. A subset of a locally convex space Y is said to be total if its span is a dense subspace of Y .

204

APPENDIX C

ClO. Let Y be a nonempty set. Afilter 9on Y is a nonempty collection of subsets of Y having the following three properties. (i) The empty set is not a member of 9. (ii) If R, A E 9, then R n A E 9. (iii) If R E 9 and A =I n, then A E 9. Now, let Y be a topological space. The collection of all neighborhoods of a nonempty subset of Y is an example of a filter. A filter is said to converge to a limit 4 E -tr if every neighborhood of 4 is a member of %.

C11. Assume that Y is a topological linear space. A filter 9 on V is called a Cauchyfilter if, for every neighborhood R of 0 in Y , there is a A E % such that A - A c R. Every convergent filter is a Cauchy filter. Y is called complete if every Cauchy filter on Y converges to a limit 4 E Y . On the other hand, a sequence {&} in Y is called a Cauchy sequence if, given any neighborhood R of 0 in Y , there exists an integer K such that f$k - 4,,,E R for all k,m > K. This condition is equivalent to the requirement that, as k and m tend to infinity independently, Y ( 4 k - 4,")+ 0 for every y in any given generating family of seminorms for.the topology of Y .Y is called sequentially complete if every Cauchy sequence in Y converges to a limit 4 E Y . The completeness of V implies its sequential completeness, but the converse is not true in general. The converse is true for metrizable spaces. A complete metrizable locally convex space is called a Frkchet space. If -tr is complete and 92 is a closed linear subspace of Y supplied with the induced topology, then 92 is also complete. C12. Let Y be a locally convex space, and r a generating family of semi4 j , where 4 j E Y ,is said to converge in Y to a norms for V . A series 4j}F= converges in Y to 4, On the other limit 4 E Y if the sequence hand, 4, is said to converge absolutely if ~ ( 4converges ~) for every y E r. An absolutely convergent series converges to some limit 4, and every rearrangement of that series converges to the same limit 4.

x;= {c:=

cj

C13. A topological linear space A whose topology is generated by a single norm is called a normed linear space. We usually denote that norm by (1 * 11 or [ l * l l A , but other symbols are also used. If, in addition, A is complete (or, equivalently, sequentially complete), A is called a (complex) Banach space. A set F c A is called nowhere dense if F has no interior points. No Banach space is equal to the union of a countable collection of nowhere dense sets. This is Baire's category theorem for Banach spaces.

205

TOPOLOGICAL LINEAR SPACES

C14. Let H be a complex linear space and let there exist a mapping (., .): {4, $}H (4, $) of H x H into C such that the following conditions are satisfied.

-

(i) (4, $) = ($, $), where the bar denotes the complex conjugate. ~ , is a linear mapping on H. (ii) For each fixed $ E H , C # J W ($) (iii) (4, 4 ) > 0 if 4 # 0. Then, (., *) is called an inner product on H. It follows that ( * , is a sesquilinear form. Moreover, with 11q511 [(4,4)]1’2,(1 is a norm on H . When H i s assigned the topology generated by this norm, it is called a (complex) inner-product space. If in this case H i s complete, it is called a (complex) Hilbert space. [A real Hilbert space is defined in the same way except that H is a real linear space and the range of (. , .) is contained in R.] a )

C15. The Schwarz inequality for the inner product (-, .) on the inner product space H states that l(4, $)] I11411 11$11. This implies that the inner product is a continuous mapping when H x His equipped with the product topology. C16. Let H be a separable Hilbert space. Then, there exists at least one sequence {ek}km, c H satisfying the following conditions.

(i) ( e k ,ek) = 1 , and ( e k ,ej) = 0 if k # j . (ii) EveryfE H can be expanded into the series m

which converges in H. When { e k }satisfies (i), it is called orthonormal and when it satisfies both (i) and (ii), it is called a complete orthonormal set. (Thus, the meaning of the adjective “ complete” in this case is different from that of Appendix Cl I .) The expansion (1) is unique in the sense that the alteration of any of the coefficients of the ek will alter the sum of the series. C17. Let { e k }be a complete orthonormal sequence in the separable Hilbert space H. Given any f E { e k } ,we have Parseval’s equation:

Conversely, the Riesz-Fischer theorem states that, for any sequence c C such that 1 akI < co, there exists a unique f E H such that c(k = (f,ek) for all k . {a,,};=

c;=

Appendix D

Continuous Linear Mappings

Throughout Appendix D, V and %’” will denote locally convex spaces, and

r and P will be generating families of seminorms for the topologies of Y and W , respectively.

D1. Continuous mappings and sequentially continuous mappings from one topological space into another topological space have been defined in Appendix B, Sections B4 and B5. If Y is metrizable, then the sequential continuity of a mapping of Y into W is equivalent to its continuity. D2. Letfbe a linear mapping of Y into 9 f . The following four assertions are equivalent. (i) f i s continuous. (ii) f i s continuous at the origin. (iii) For every continuous seminorm p on W , there exists a continuous seminorm y on Y such that p ( f ( 4 ) ) I y(4) for all 4 E V [or, equivalently, y ( 4 ) < I implies that p ( f ( 4 ) ) I I]. 206

207

CONTINUOUS LINEAR MAPPINGS

(iv) For every p E P, there exist a constant M > 0 and a finite collection {yl, . . . , )7, c r such that

for all

4 EY.

D3. The spaces Y and W are said to be isomorphic if there exists a bijection I of Y onto *W such that I is continuous and linear and its inverse i-' is also continuous and linear. In this case, I is called an isomorphism of Y onto W . If Y and W are the same space, then i is called an automorphism on Y . D4. If Y is separated, there exists a complete locally convex space and a mapping I of V into 9 such that the following three conditions are satisfied. I is an isomorphism of Y onto the image of Y in 9 supplied with the topology induced by 9. (ii) The image of Y is dense in 9. (iii) Given any complete separated topological linear space W and any continuous linear mappingfof Y into W , there exists a unique continuous linear mapping f o f 9 into W such thatfis equal to the composite mapping f i obtained by first applying I and then applying

(9

f. Moreover, if 4 is any other complete locally convex space for which there exists a mapping 7 satisfying conditions (i) and (ii), then .lr" and 9 are isomorphic. The space 9 is called the completion of Y . D5. Let Y be a Frkchet space, % a dense linear subspace of V supplied with the induced topology, W a sequentially complete separated space, and f a continuous linear mapping of % into W . Then, there exists a unique continuous linear mapping g of Y into W such that g ( 4 ) =f(+) for all

4€%. A similar result for sesquilinear forms is the following. Let W be as before, let Yl and Y 2be Frechet spaces, and let 421 and a2be dense linear subspaces of Yl and Y 2respectively. Supply Y , x Y 2with the product topology and 4Yl x %2 with the induced topology. Assume that f i s a continuous sesquilinear mapping of a1x 4Y2 into W . The continuity property is equivalent to the condition that, given any p E P, there is a constant M > 0 and two continuous seminorms y1 and y 2 o n llr, and V 2 ,respectively, for which

P [ f ( A *)I

&1(4)Y2(*)?

4 E @l?

*

E @2.

(1)

208

APPENDIX D

We can conclude that there exists a unique continuous sesquilinear mapping g of Y l x Y , into W such that g ( 4 , $) =f (4, $) for all 4 E 42, and I) E 42,. Moreover, (1) holds again for f replaced by g and for all 4 E Y , and $ E Y , .

D6. Let f be a linear mapping of Y into W .f is called a bounded mapping if it takes bounded sets in Y into bounded sets in W . Iff is continuous, it is bounded, but the converse is not true in general. However, the converse is true when Y is a metrizable space. D7. A functional on Y or a form on Y is a mapping of Y into C. The space of all continuous linear functionals on Y is denoted by Y ' or by [ Y ;C ] and is called the dual of Y . Given any f E Y ' ,the mapping yf: 4If (4)I is a seminorm on Y .The topology generated on Y by the collection {y,},.,. of ail such seminorms is called the weak topology of Y . If Y is separated and 4 is any member of Y such that 4 # 0, there exists an f E Y ' such that f(4)# 0. Thus, under its weak topology, Y is again a separated locally convex space. Moreover, if A is a Banach space and {ak}is a sequence that converges under the weak topology of A , then supkIlakII < co.

D8. The symbol [ Y ;W ]denotes the set of all continuous linear mappings of Y into W . The addition and multiplication by a complex number of members of [ Y ;W ] is defined as in Appendix A5. As a consequence, [ Y ;W ] is a linear space. Given any bounded set R c Y and any p E P, the mapping

fH4.n SUP P(f(4k

vr2, p :

f E [-tr ; *Wl

defines a seminorm on [ Y ;W ] .The collection of all such seminorms {vr2,p } , where R traverses the bounded sets in Y and p traverses P, defines a locally convex topology on Y , which is called the topology of uniform convergence on the bounded sets of Y or simply the bounded topology. On the other hand, when R traverses only the finite sets in Y and p traverses P, the topology generated by {qn, p } is called the topology of pointwise convergence on Y or simply the pointwise topology. The pointwise topology is weaker than the bounded topology. Unless the opposite is explicitly stated, [ Y ;W ] is understood to have the bounded topology. When it has the pointwise topology, it is denoted by [ V ;WJ".

D9. A subs$ B of [ Y ;W ] is said to be equicontinuous if, given any p E P, there exists a constant M > 0 and a finite collection {yl, . . . , y,} c I' such that

p(f(4)) 5

f €I

max Yk(4)

1SkSm

CONTINUOUS LINEAR MAPPINGS

209

for all ~ E YIf .Y is a FrCchet space, the following three assertions are equivalent : (i) E is bounded in [ Y ;W ] . (ii) E is bounded in [ Y ;WIu. (iii) Z is equicontinuous. D10. The following is one version of the closedgraph theorem: Let Y and W be FrCchet spaces and f a linear mapping of Y into W . Assume that, for every sequence { c $ ~ which } converges to zero in Y and for whichf(4,) converges to t,b in W , we always have that I) = 0. Then, f is continuous.

D11. Let A and B be normed linear spaces with norms I I . I I A and l l * l l B , respectively. The bounded topology of [ A ; B ] is equal to the topology generawhere ted by the single norm 11. I([A;

(The supremum notation means that the supremum is taken over all a E A for which llallA = I.) This topology is called the uniform operator topology. When B is a Banach space, [ A ; B ] under the uniform operator topology is a Banach space, too. For any fixed a E A , the mapping y o : f H 11 f(a)llB is a seminorm on [ A ; B ] . The topology generated by thecollection { y a : a E A } of seminorms is called the strong operator topology of [ A ; B ] . It is the same as the pointwise topology of [ A ; B ] , defined in Appendix D8. In this case, if A is a Banach space, [ A ; B ] under the strong operator topology is a sequentially complete separated space; if, in addition, {fk}km, converges in this topology tof, then llfll I h~+mllhll. Finally, for any given a E A and b‘ E B’ (8’ is the dual of B), Y ~ , ~ , : ~ H Ib‘(f(a))I is also a seminorm on [ A ; B ] . The topology generated by the collection { y o , 6 , : a E A , 6‘ E B’} of seminorms is called the nieak operator topology of [ A ; B ] . Each of these topologies separates [ A ; B ] . D12. Upon applying Appendix D9 in the context of Appendix DI I , we obtain the principle of uniform boundedness: If E is a subset of [ A ; B ] and if ~~p,,,llf(~ ~~ <)co for~ each a E A , then s ~ p ~ . . ( ( f ( l [<~co. ;~~ Upon combining this principle with the next paragraph we get the following < co for each a’ E A‘, then result: Tf R is a subset of A and if supaEn)a’(a)( SUPaEnllallA < co.

D13. The second dual A” of a Banach space A is the dual of the dual A’ of A . Every U E A generates a functional a” on A’ through the definition (a”, 6 ) A ( 6 , a), where b E A’. It follows that a” E A ” and that lla”ll = Ilall.

210

APPENDIX D

D14. Let A and B be Banach spaces. I f f € [ A ; B ] is bijective (so that its inversef-’ exists and B is the domain off-’), thenf-’ E [ B ;A ] .

D15. Let H be a Hilbert space. Corresponding to each g E H ,where H’ is the dual of H , there exists a unique b E H such that g(a) = (a, b) for every a E H ; the converse is also true. Moreover, lIgll[H;C] = / / ~ I / H . Next, let f be a sequilinear form on H x H , which is continuous when H x H is supplied the product topology [i.e., there exists a constant M for which I f ( a , b)l I Mllall llbll for all a, b E HI. Then, there exists a unique P E [ H ; HI such that (Pa, 6) = f ( a , 6) for all a, 6 E H. An operator P mapping H into H is called positive if (Pa, a) 2 0 for all a E H . The set of positive continuous linear mappings of H into H i s denoted by [ H ;H I , . Similarly, a sesquilinear f o r m f o n H x H is called positive if f ( a , a ) 2 0 for all a E H . Iff is a positive sesquilinear form on H x H such that f ( a , a) I Mllall’, where M is a constant not depending on a, then there exists a unique P E [ H ; H I , such that (Pa, b) = f ( a , 6) for all a, b E H . Furthermore, if Q E [ H ; J ] , where J is another Hilbert space, its adjoint Q’ is defined by (Q’a, 6) (a, Qb), where a E J and b E H . We have that Q’ E [ J ; HI and IIQ’II = IIQII. A Q,E [ H ; HI is called self-adjoinf if Q = Q’; this is the case if and only if (Qa, a) is real for all a E H. For such an operator, we have

Q E [ H ; HI is called skeliT-adjoint if Q self-adjoint.

t

=

- Q’; this is so if and only if iQ is

Appendix E

Inductive-Limit Spaces

El. Let N be any index set. Also, let Y be a linear space such that Y = UnENYn where , each Y , is a locally convex space. We define an inductive-limit topology on Y by assigning the following base W of neighbor. if and only if A is a balanced convex absorbent set in hoods of 0 to V :A E @ V and, for every n E N , A n Y , is a neighborhood of 0 in Y ,. In this case, Y is a locally convex space. It is called an inductive-limit space or the inductive limit of {Y,}.Moreover, the collection d of all convex subsets of Y such that each A E d intersects every Y , in a neighborhood of 0 in Y , is also a base of neighborhoods of 0 in Y . It is convenient to allow the following trivial situation as a special case of an inductive-limit space. Every locally convex space Y can be considered to be the inductive limit of itself by letting N be any index set and setting Y , = Y for every n E N .

E2. Let Y = U n e N Y , be an inductive-limit space, let W be a locally convex space, and let f be a linear mapping of Y into W .Then,fis continuous if and only if, for every n E N , the restriction o f f t o Y , is continuous. 211

212

APPENDIX E

In particular, if each Y , is a metrizable locally convex space and if the restriction off to each Y , is a bounded mapping, then f is continuous on Y .

E3. Let Y be a linear space such that Y = the following three conditions hold :

u."=

Y , , where, for each n,

(i) Each Y , is a locally convex space. (ii) Y , c Yn+l. (iii) The topology of Y , is equal to the topology induced on Y , by Yn+1.

Finally, let Y have the inductive-limit topology. Under these conditions, Y is called a strict inductive-limit space or the strict inductive limit of {Y,}r= As a consequence of this definition, the topology of every Y , is equal to the topology induced on Y , by Y . Moreover, if each Y , is complete and separated, then so, too, is Y .

E4. Let Y be the strict inductive limit of {Y,}."_,. If, for each n, Y , is a Y is said to possess the closure property. Under this closed subset of Yn+l, circumstance, a subset R of Y is bounded in Y if and only if, for some n, R c Y , and R is bounded in Y,'.Also, a sequence converges in Y if and only ifit is contained and converges in Y , for some n. It is a fact that Y possesses the closure property if every Y , is a complete separated space.

Appendix F

Bilinear Mappings and Tensor Products

F1. Let %, Y , and W be topological linear spaces. A bilinear or sesquilinear mapping f on % x Y into W is called separately continuous if 4t+ f ( 4 ,$) is continuous on @ for each fixed @ E Y and $ ~ (4,f$) is continuous on Y for each fixed 4 E %. I f f is continuous from @ x Y into W when % x Y is supplied with the product topology, then f is separately continuous.

F2. Let % be a metrizable locally convex space or the inductive limit of such spaces, and let Y be a Frtchet space or the inductive limit of FrCchet spaces. Assume that f is a separately continuous bilinear or sesquilinear mapping of 42 x Y into a locally convex space W . Then, f is continuous on % x V when @ x Y is equipped with its product topology. Let % and Y be FrCchet spaces. The topologies of % and Y can always be obtained from two multinorms {yn}F= and {[,},"= respectively having the and < C2 < * * Assume monotonic properties: y1 < y z 5 y 3 <.

-

213

c3

a .

214

APPENDIX F

that W is a complex Banach space B. Then, there exist a nonnegative integer rn and a constant M > 0 such that

IIf(4, $)lls for all

MYrn(4Krn($)

4 E 42 and $ E Y .

F3. In this paragraph, 42 and Y are complex linear spaces without any topologies. f denotes an arbitrary bilinear form on % x Y ;i.e., f E B(%, Y ) . For any given 4 E oa and $ E Y ,the mappingfwf(4, $) is a linear form on B(%, Y ) . We denote this mapping by 4 @ $ and therefore have 4 0 $ E B(%, Y ) * .It follows easily that the operator X: {$, $}H 4 @ $ is a bilinear mapping of % x Y into B(%, Y ) * .The span of X(42 x Y ) is denoted by % @Y and is called the tensor product of % and Y . Also, X is called the canonical bilinear mapping of % x Y into % @ Y . Every element 0 E % 0 Y has a nonunique representation of the form 0 = xi, 4 k @ $ k , where 4 k E % and $k E Y . F4. Now, let q,Y , and W be three complex linear spaces without topologies. I.(% @ Y , W ) denotes the ,linear space of all linear mappings of % @ Y into W , and B(42, Y ;W ) denotes the linear space of all bilinear mappings of % x Y into W . Given any h E L ( %@ Y , W ) ,hX denotes the composite mapping on 42 x Y into Y obtained by first applying X and then applying h. We can now state a theorem: The mapping h w h X is a linear bijection of L(% 0 V ,W ) onto B(%, V ;W ) . The significance of this theorem is the following. Tensor products allow us to replace linear spaces of bilinear mappings by linear spaces of linear mappings. This is one motivation for introducing tensor products.

F5. Let 42 and Y be locally convex spaces and let r and H be bases of continuous seminorms for % and Y , respectively. Given any y E r and q E H, define the function p o n any 0 E % 8 Y by

That is, the infimum is taken over all representations of 0 of the form 0 = Ckc j k @ $ k . Then, p is a seminorm on 42 @ Y ,and p ( 4 @ $) = y(4)q($) for all 4 E % and $ E Y .Also, p is a norm if and only if both y and q are norms. The collection P of all such p is taken to be a generating family of seminorms for a topology 0, on % B Y , called the projective tensorproduct topology or the n-topology. In fact, P is a base of continuous seminorms for 0,. 8, is the strongest locally convex topology on % @ Y under which the canonical bilinear mapping X defined in Appendix F3 is continuous.

BILINEAR MAPPINGS AND TENSOR PRODUCTS

215

Also, 42 0 Y is separated if and only if both and Y are separated. Henceforth, it is understood that @ @ Y is supplied with the topology 0,.

F6. It is a fact that a linear mapping h of 42 0 Y into W is continuous if and only if the bilinear mapping hX of 42 x Y into W is continuous. (Here, @ x Y is supplied with the product topology.)

F7. Let 42 and Y be normed linear spaces with norms y and q, respectively. Then, 42 0 Y is a normed linear space whose norm p is defined on any 6 E 42 0 Y by (I). The completion of @ 0 Y (see Appendix D4) is denoted by 42 6 Y and is the collection of all equivalence classes (equivalence taken in the sense of Cauchy sequences under the norm p ) of series of the form 1 6 k 0 $k where 6 k E @, $k E Y , and 3

m

7(4k)d$k) k= 1

< 03.

Thus, @ 6 Y is a Banach space. Its norm p is given on any 0 E @ @ Y by

5

de>= inf(k = 1y ( 6 k ) d $ k ) :

m

=

6 k k= 1

0 $k].

Appendix G

The Bochner Integral

G1. Let T be a nonvoid set. A nonvoid collection (5 of subsets of T is called a a-algebra of subsets of T or simply a a-algebra in T if the following two conditions hold : (i) If E E (5, then T\E E (5. (ii) Ek E (5 if every Ek E (5.

ukm,,

It follows that T and the empty set are both members of (5, and that E (5 E (5. Also, R\A E (5 if R, A E (5. The members of (5

n=; Ek if every Ek are called measurable sets.

62. Given any collection G of subsets of T, there exists a smallest a-algebra (5 of subsets of T such that 6 c (5. (5 is said to be the a-algebra generated by 6.Now, assume that T is a topological space. Let (IB be the a-algebra generated by the collection of all open sets in T. The members of (5, are called the Bore1 sets in T .

G3. Let T and X be two nonvoid sets and let (5 and (5’ be a-algebras of subsets of T and X , respectively. The product a-algebra in T x X is the 216

THE BOCHNER INTEGRAL

217

smallest a-algebra of subsets of the Cartesian product T x X that contains . a-algebra is every set of the form E x E', where E E 6 and E ' E ~ 'This denoted by 6 x (5'. (Thus, in this case, (5 x 6' does not represent the Cartesian product of (5 and K'.) 64. A positive measure p (or, on the other hand, a complex measure) is a mapping of a a-algebra 6 in T whose range is in [0, 001 c Re1(or, respectively, in C) and which is a-additive. The last phrase is defined as follows. p is called a-additive on (5 if, for every sequence {Ek}km, of pairwise disjoint sets Ek E 6, we have that p(Uk &) = x k p(Ek).However, if this condition is required to hold only for finite collections { E k } i = l ,then p is called additive on 6. p is a-additiveon6ifandonlyifitisadditiveon (5 and p(UpE1Ek) = limk+mp(&) for every increasing sequence {Ek};=l in 6 (i.e., Ek E 6 and Ek c Ek+l for every k). A positive measure p is calledfinite if p(T) co. Throughout Appendix G, p will be allowed to be either a positive or complex measure unless it is explicitly restricted to being just one of these types of measures, and we shall refer to p as a measure.

-=

6 5 . An important special case is Lebesgue measure on the Borel subsets of R". This is the measure that assigns to each interval [x, y ] A {t E R": x 5 t 5 y} the value vol[x, y ] A 1(yk - xk), where x = {xk}i= E R" and y = {yk}i=l E R". Lebesgue measure is a positive measure and is unique in the following sense. No other positive measure on the Borel subsets of R" assigns the value vol[x, y ] to every interval [x, y ] c R".

66. A set N E 6 such that p ( N ) = 0 is called a p-null set and is said to be of measure zero. Let P denote a property that each point t E E E (5 either has or does not have depending on the choice of t. We say that P holds almost everywhere on E or for almost all t E E when there exists a p-null set N such that P holds for every point of E\N. G7. A partition 7c of E E 6 is a finite collection {Ek};= of pairwise disjoint sets Ek E 6 such that E = Ur= E k . Let p be a complex measure on (5. We define a function 1p1 on (5 into [0, 001 by IpI(E)Asup n

f.

Ip(Ek)lr

EE(5,

k=l

where the supremum is taken over all partitions 7c of E. It is a fact that 1p1 is a finite positive measure on 6. Moreover, I p ( E ) I 5 I p I ( E ) 5 1 p 1 (T) < co for every E E ~ lpl . is called the total-variation measure for p, and IpI(T) Var p is called the total variation of p on T. When the range of p happens to be real and contained in [0, co), we have that 1p1 = p.

218

APPENDIX G

Now, let p be any positive measure on (5. Thus, p is now allowed to assign co to some of the members of (5. I n this case, we set IpI p.

GS. Let S be any subset of T. The characteristic function xs of S is the function on T defined by xs(t) = 1 if t E S and X s ( t ) = 0 if t $ S. Let A be a complex Banach space. An A-ualued simple function f on T is any function of the form f = akXEk, where ak E A and {Ek};=, is a partition of T. The simple function f is said to be integrable or p-integrable if ak = 0 whenever I p I (Ek) = co. The Bochner integral qf f on T is denoted by -7f A j T f dp A j T dFfand defined to be akp(Ek), where now we use the convention that, if ak is the zero element 0 E A and p(&) = co,then the product Oco is the zero element 0 E A. G9. A function f on T into A is called measurable or p-measurable if, for any E E (5 such that I p I ( E ) # co, there exists a sequence of A-valued simple functions on T that converges pointwise to f ( t ) for almost all t E E. I n this case, (1 f ( . ) l l , is also measurable as a function on T into R . In the special case where T = R", we have the following result. Every weakly continuous function on R" [i.e., every function f from R" into A such that a'f(-) is continuous for all a' E A'., where A' is the dual of A] is measurable. G10. Let {fk}km, be a sequence of integrable A-valued simple functions on T such that

jTlif,c(*)

-fj(*>IIA

-to

~ I P I

(1)

as k a n d j tend to co independently. It is a fact that there exists an A-valued functionfon Tand a subsequence {Ak} of { f k } such that h k ( f ) -+ f ( t ) for almost all t E T. There will be other such functions and subsequences. However, it is also a fact that g is another such function if and only if f ( t ) = g ( t ) for almost all t E T. Thus, the sequence {h}determines a class F consisting of all functions each of which differs from f on no more than a p-null set. Still more is true. The sequence {3fk}F=, of Bochner integrals of the f k converges in A . Its limit is called the Bochner integral 3f of f on T with respect to p, wherefis any member of F. We also use the notation 3 f A j T f dp jT dpf and say that f i s integrable or p-integrable on T. The subscript Tmay be dropped when this leads to no confusion. The notation

3f

jm 4 4

p p ,f ( t )

is used when it is useful to display the independent variable of the functionf. When Tis a Borel subset of R" and p is Lebesgue measure on the Borel subsets of T, we write 3 f 4 j T f ( t )dt.

THE BOCHNER INTEGRAL

219

Iff and g are both members of the same class F (i.e., if they differ on no more than a p-null set, then 3f= 3g. Furthermore, if {gk}is any other sequence of integrable A-valued simple functions on T satisfying JTllgk(*)-gj(')llA dip( --*O

as k , j + co independently and determiningfas above, then {3gk} also converges to 3f. Thus, -7fis independent of the choice of the sequence {fk}. Finally, we mention that a function is p-integrable if and only if it is I p I -integrable.

G11. Iff is a pintegrable A-valued function on T, then llf(*)llA is a IpIintegrable nonnegative function on T. Moreover, we have the useful estimate I I m 4

5

JTllf(*)llA

dlpl.

The right-hand side remains the same for every memberfof a given class F.

G12. Let 9 denote the space of all complex-valued smooth functions on R" of compact support. Also, let v be a function that assigns a complex number to each bounded Borel set in R" in such a way that, for each sphere S, A { t E R": I tI I m } , where m = 1, 2 , . . . , we have that v is a complex measure on the a-algebra of all Borel subsets of S,. Such a function v is called a a-jinite complex measure. Given any 4 €9,choose m such that supp 4 c S, and set j R n4 dv A js, 4 dv. Then, 4 dv exists for each 4 € 9and is independent of the choice of S,. Moreover, a knowledge of the values of jR" 4 dv for every 4 € 9uniquely determines the values that v assigns to all the bounded Borel sets in R". (See Schwartz, 1966, p. 25.) Similarly, if both g .and h are v-integrable A-valued functions on S, for whenever 4 ~ 9 We . again set every m , then so, too, are g4 and SRng4 dv A ss,,g4 dv, where S, is chosen for the given 4 as before. If J R n g4 dv = j R nh 4 dv for all 4 €9, then g(t) = h(t) for almost all t E R".

sRn

G13. Given T, (5, p , and A, the set of all integrable A-valued functionsfon Tcan be partitioned into equivalence classes F as indicated in Appendix G10. I f f and g are members of the same F, then j f d p = s g dp and, in addition, Ilf(.>ll dlpl llg(.>lld l p l . We define F d p A s f d p , where f is any member of F. The notation L,(T, (5; p ; A) = Ll(p; A) denotes the linear space of all equivalence classes F, and this space is a Banach space under the norm 11 * where

s

=s

s

IIFIIL,

'J

T

ll.f(*N4~lPl9

f E F .

220

APPENDIX G

When T is R",6 the collection of all Bore1 subsets of R",and p Lebesgue measure, we denote L,(p, A) simply by &(A), and L,(C) by L,. It is customary to represent F by any member f E F and to replace F by f in all manipulations on the members F of L l ( p ; A ) . In fact, we shall say that the members of L l ( p ; A ) are the functions f and will maintain the tacit understanding that f should really be replaced by the equivalence class F to which f belongs. 614. Let 9(T,6 ;C ) be the space of all complex-valued functions g on T that are the limits under the norm ( 1 . JIG, where llgllG supfETIg(r)I,of sequences of simple functions on T. Thus, every member of 9(T, 6 ;C ) is a bounded function. Iff E L,(T, 6 ;p ; A ) and g E Q(T,6 ;C ) , then f g E L,(T, 6 ; p ;A ) . Furthermore, if T, X , 6,and 6' are as in Appendix G3 and if I E G(T x X , 6 x 6';C ) , then, for each fixed x E X , we have I(-, x) E G(T, 6 ;C). G15. I f f is a p-measurable function on T into A and if )If(?)[IA where g E L,(T, 6 ;p ; R),then f E L,(T, 6 ;p ; A ) .

Ig(t),

616. Let Llo(T,6 ;p ; A ) = L l 0 ( p ;A ) denote the linear space of all integrable A-valued simple functions on T. [Here again, it is understood that the members of L l 0 ( p ;A) are really equivalence classes of functions differing ffom the simple functions on no more than p-null sets.J Given anyf E L l ( p ; A), we can choose a sequence {hk}km,1c L I o ( p ;A ) such that hk + f i n Ll(p; A ) and hk(t) -+ f ( t ) almost everywhere on T. (Any one of the subsequences mentioned in the first paragraph of Appendix G I 0 will do.) Consequently, L l 0 ( p ;A ) is dense in L l ( p ; A ) . Moreover, L , ( p ; A ) is the completion of Ll0( p ;A ) under the norm 11 [I L,.

G17. Let A4 be a continuous linear mapping of thecomplexBanach spaceA into another such space B. Iff € L 1 ( T ,6 ;p ; A ) , then M f ( * )E L ~ ( T6,;p ; B) and

j f& In particular, i f f dual of B, then

E L,(T, 6 ;p ;

=

JWv) 4.4

*

[ A ; B J ) , a E A , and 6'

E

B', where B' is the

,

G18. The following is the theorem of dominated convergence. If {fk}p= c L l ( p ; A ) , if Ilfk(t)ll Ig(t) for each k,where g E L 1 ( p ;R ) , and iffk(t) +f ( t ) for

221

THE BOCHNER INTEGRAL

-

almost all t E T, then f E L , ( p ; A) and, as k + and JT h dp $T f dp.

(319. Let p E R and p 2 1. Let {fk}:= simple functions on T such that slIfk(')

03,

JT

Ilh(t)-f(t)II

dlptl - 0

be a sequence of integrable A-valued

+o

-fj(')IIAp

as k and j tend to 00 independently. Then, there exists an equivalence class F consisting of all A-valued measurable functions f on T such that any two members of F are equal to each other for almost all t E F and, for any f E F, the integral

s

Ilfk(

exists and tends to zero as k + and

s Ilf(

')//Ap

'

)-f(

00.

=

*

)IIAp

dl p1

Moreover,

lim

k-tm

1

Ilfk(

11 f ( ' ) l l A P ')IIAp

E

Ll(T, 6 ;lpl ;R),

dlpl.

The space of all such equivalence classes F is denoted by L,(T, 6 ;p ; A ) = L p ( p ;A ) , and it is a Banach space under the norm 11. [IL,, where

As before, we shall represent F by any one of its membersf. Moreover, we shall speak of L p ( p ;A ) as consisting of all f in all F, the partitioning of the space of all such f into equivalence classes being understood. It is a fact that f E L p ( p ;A) if and only i f f is an A-valued measurable function of R" and IIf ( . ) I I A E Lp(p;R). In the special case where is the a-algebra of Bore1 subsets of R" and p is Lebesgue measure, we denote Lp(p;A) by L,(A) and Lp(C)by L,.

G20. The following is an extension of Holder's inequality. Let p , q E R be such that p > 1 and p - ' + 4 - l = 1. Moreover, let f E L,(T, 6 ;p ; A) and g EL&T,6 ;p ; C ) . Then,fg E L , ( T ,6 ;p ; A ) , and A

References

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Index of Symbols

A.4 W,l64 B,4 B('lI, j"; tr), 196 91(B),10 7 18= 18lJl' 153 ~=~lJl' 153 18p,43

Q:.(A),5 0 ~+

meA), 67

e: m(A), 67
15,

G(A),24 25 Q:; C),220 ~(T x X, Q: x '1:'; C),4O ~,(H) = fI,(T, Q:; H),46 ~o(A) = ~o(T, Q:; A), 24 ~(A), ~(T,

en,3 C+,131 ~Km(A), 57 '1:,23 '1:"",29 diam, 4 D"14 ~(A), 50 ~J[(A), 50 ~(A), 65 ~J[m(A), 50 ~Lp (A), 72

H,4 .;tf',81 inf,4 1,145 J(A),65 J"'(A),6 5 JJ"'(A), 64

lim, 4 lim, 4 225

226

INDEX OF SYMBOLS

L 1(JL ; A) = L 1(T, Q:; JL; A), 219 L 1 0 (JL ; A) = L 1 O(T, Q:; JL; A), 220 L~oC (H), 150 L,(JL; A) = Lp(T, Q:; JL; A), 221 !l'c. d(A), 1I8 !l'~ a(A), 66 !l'(w, z; A), 1I8 !l'm(w, z; A), 66 2, 1I8 max, 4 min,4 %[T], 147 9/,78 .P,23 R+,3 R",3 Re",3 9l[Tj, 147

SSVar,24 sup, 4 supp,4,55 SVar,24

"p.,,82 pJ,p,64

a,,54

0,2

0.,2

Some Additional Special Symbols

1+, 58, 73

MI., 12 ~,

4

0,4 .,90

*,97 -,155 0,37,61,69

6,37 (8),214 @,215 ,." 10, 14, 24 [·j,4

11'11 = II'IIA, 204 11'11.. 37

.'I'm(A),66

(., '),4, 205

Var,24 vol, 4 :!E, 156

(', 'j, 4 [-, '),4 'j, 4 [.; 'j, 3, 208 [.; 'j., 68

Yc,d.t.

r.

1I8

yo, 50 Yp. 0, 72

[.; -r. 3, 68, 208

Ye,53,68 SJ. 0, 57 0.,12 K c • d, 66, 1I8

'1-,2 ·.....·,2 'H',2

<', '),3

A

Banach space, 204 Banach system, 79 Base of continuous seminorms, 203 Base of neighborhoods, 199 Beltrami, E. J., 130, 131 Berberian, S. K., 147, 190, 193 Bijection, 3 Bilinear form, 196 Bilinear mapping, 196 Binomial coefficient, 15 Black box approach, xi Bochner integral, 218 Bochner-Schwartz theorem, 156 Bogdanowicz, W., 81 Bore1 set, 216 Boundary of a set, 198 Bounded function, 5, 208 Bounded' function, 131 Bounded*-real function, 138 Bounded set, 203 Bounded topology, 53,208

[A; B]-valued distribution, 52 Absolutely convergent series, 204 Absolutely convex set, 196 Absorbent set, 196 Additive set function, 24 Adjoint operator, 210 Admittance operator, 80 Akhiezer, N. I., 178 Algebraic dual, 196 Analytic function, 17-1 9 strongly, 18 weakly, 17-18 Antilinear mapping, 197 Automorphism, 207

B &valued distribution, 52 Baire's category theorem, 204 Balanced set, 196 227

228

INDEX

C

Canonical bilinear mapping, 214 Canonical injection, 200 Carlin, H. J., 80, 137, 144, 146, 147, 151 Cartesian product, 2, 200 Castriota, L.J., 137, 151 Cauchy filter, 204 Cauchy sequence, 204 Cauchy’s integral formula, 21 Cauchy’s theorem, 20 Causality, 93 Change-of-variable formula, 111-1 12 Characteristic function, 21 8 Cioranescu, I., 92 Closed graph theorem, 209 Closed set, 198 Closure, 198 Closure property, 212 Compact set, 4 Complete orthonormal set, 205 Complete space, 204 Completion of a space, 207 Complex conjugate .of an operator, 138 Complexification, 137 Complex measure, 217 Composition operator, 90 Composition product, 90 Conditions E, 97 Constant distribution, 108 Continuous mapping, 199 Contour integration, 20-22 Contractive operator, 129 Convergent filter, 204 Convergent sequence, 199-200 Convex hull, 196 Convex set, 196 Convolution operator, 98, 112-1 15 commutativity with shifting and differentiation, 104 Convolution product, 97 Copson, E. T., 20 Counting measure, 40 Cristescu, R.,92

D DAmato, L., 144, 192 Delta function, 55 Dense set, 198

Dependent variable, 2 Diameter of a set, 4 Differentiation, 12-17 generalized, 54, 68 Dinculeanu, N., 24 Direct product, 110 Distribution, 52 generated by a measure, 56 independent of certain coordinates, 110-111 Lp-type, 75 regular, 56 Dolezal, V.,92 Domain, 2 Dominated convergence, theorem of, 220-221 Dual space, 208 Dunford, N., 33, 194

E Ehrenpreis, L.,81 Equicontinuous set of mappings, 208 Euclidean space, 3 Exchange formula, 119

F Filter, 204 Finite-dimensionally-ranging function, 37 Finite-dimensional subspace, 195 Finite measure, 217 Foias, C., 141 Form, 208 Fourier transform, 155 FrCchet space, 204 Function, 3 Functional, 208

G Gask, H., 81 Gelfand, I. M.,81, 156 Generalized differentiation, 54, 68 Generalized function, 67 [A; B]-valued, 67 B-valued, 67

INDEX

Generating family of seminorms, 203 Giordano, A. B.,80, 144, 146, 147 Glazman, I. M., 178 Graph, 2 Gross, B.,xi

229

Kernel representation, 91 Kernel theorem, 85 Konig, H., 150, 153, 187 Kritt, B.,174 Kurepa, S., 180

L H

Hackenbroch, W., 23,29, 150, 153, 157, 164, 174, 175 Hausdorff space, 199 Hilbert n-port, 143 Hilbert port, 78, 80 Hilbert space, 205 Hille. E., 194 Holder's inequality, 221 Horvath, J., 194

I Imaginary part of an operator, 138 Impedance operator, 81 Improper integral, 11 Independent variable, 2 Induced topology, 200 Inductivslimit space, 211 m-port, 79 Injection, 3 Inner product, 205 Integrable function, 30 Integration by parts, 17 Interior point, 198 Interval in R",4 Isometric operator, 140 Isomorphism, 207

J Jones, D. S., 78

K Kaplan, W., 112 Kernel of %, 91 Kernel operator, 90

&-type distribution, 75, 125-128 L,-type testing function, 72 Laplace transform, 118 strip of definition for, 118 Laplace-transformabledistribution, 118 Lebesgue measure, 217 Leibniz's rule, 15 Linear combination, 195 Linear form, 196 Linearly independent set, 196 Linear mapping, 196 Linear space, 194 Locally convex space, 202. Locally essentially bounded function, 56 Local mapping, 157 Lossless Hilbert port, 139-143 Lossless operator, 140 Lossless scattering transform, 145 Love, E. R., xi

M McMillan, B., xi Mapping, 3 Marinescu, G., 92 Matched port, 146 Measurable function, 218 Measurable set, 216 Measure, 217 Meidan, R., 80, 92 Meixner, J., xi, 150 Metric space, 200 Metrizable space, 200 Multinorm, 203

N Neighborhood, 199 Nested closed cover, 64 Newcomb, R. W., xi, 137

230

INDEX

Norm, 202 Normal space, 65 Normed linear space, 204 Nowhere dense set, 204 Null set of a distribution, 55

0

One-to-one mapping, 3 Open set, 198 Operator, 3 Operator-valued measure, 26 Order of a derivative, 14 Orthonormal set, 205

P Parseval’s equation, 136, 155, 205 Partition, 217 Passive operator, 150 Phillips, R. S., 194 dopology, 214 Pointwise topology, 53, 68, 208 Polarization identity, 197 Pondelicek, B., 92 Positive-definite distribution, 156, 174 Positive* mapping, 178 Positive measure 217 Positive operator, 210 PO measure, 26 Positive*-real mapping, 181 Positive sesquilinear form, 160-168, 197, 210 Primitive, 108 Principle of uniform boundedness, 209 Product u-algebra, 216-217 Product topology, 200 Projection, 58

R Range, 2 Real convolution operator, 138 Real distribution, 138 Real linear space, 197 Real part of an operator, 138

Real signal, 138 Realizability theory, xi Reciprocal scattering transform, 145 Rectangular partition, 6 Refinement of a partition, 6 Regular distribution, 56 Regularization, 106-108 Relation, 2 p-type testing function space, 65 degenerate, 65 Riemann integration, 6-1 1 Riemann sum, 6 Riesz-Fischer theorem, 205 Robertson, A. P., 194 Robertson, W., 194 Rudin, W., 33, 40, 179, 194

S 6-set, 65 6;-topology, 53, 68 Sabac, M., 92 Scalar semivariation, 24 Scattering operator, 80 Scatter-passive operator, 129 Scatter-semipassive operator, 129 Schaeffer, H. H., 194 Schwarz inequality, 197, 205 Schwartz, J. T., 33, 194 Schwartz, L., 49, 81,92, 110, 128, 156, 219 Schwindt, R., 178 Sebastiao e Silva, J., 57 Second dual space, 209 Self-adjoint operator, 210 Seminorm, 202 Semipassive operator, 150 Semivariation, 24 Separable space, 198 Separated space, 198-199 Separately continuous bilinear mapping, 213 Sequentially complete space, 204 Sequentially continuous mapping, 200 Sequentially dense set, 200 Sesquilinear form, 197 Sesquilinear mapping, 197 Shifting operator, 54 o-additive measure, 217 u-additive set function, 26

231

INDEX

a-algebra, 216 a-finite complex measure, 219 a-finite PO measure, 30 Simple function, 24, 218 Skew-adjoint operator, 210 Smooth function, 14 Span, 195 Strict inductive-limit space, 212 Strongly measurable function, 56 Strong operator topology, 209 Subspace, 195 Support, 4 of a distribution, 55 Surjection, 3 System function, 114 Sz.-Nagy, B., 141

U Uniform continuity, 5, 97 Uniform convergence, 9 Uniform operator topology, 209 Unitary operator, 142 Unit impulse, 114 Unit-impulse response, 114

V Vilenkin, N. Ya., 81, 156 Vladimirov, V .S.,186 Volume, 4

T

Tempered complex measure, 157 Tempered distribution, generated by a PO measure, 163 Tempered positive measure, 156 Tempered PO measure, 162 Tensor product, 214 Testing functions of rapid descent, 66 Time invariance, 112 Time-varying operator, 112 Toll, J. S., xi Topological linear space, 201 Topological space, 198 Topology, 198 Topology of uniform convergence on @-sets, 68 Total subset, 203 Total variation, 24, 217 Transformation, 2 Translation-invariance, 1I2 Translation-invariant sesquilinear form, 161 Translation operator, 54 Translation-varying operator, 112 Treves, F., 194

W Weak operator topology, 209 Weak topology, 208 Wexler, D., 90 Willems, J. C., 157 Williamson, J. H.,136 Wohlers, M. R., xi, 130, 131 Wu, T. T., xi

Y Youla, D. C., 137, 151

L

Zaanen, A. C., 178, 194 Zemanian, A. H., 55, 62, 66, 79, 83, 92, 108, 114, 117, 119, 121, 131, 137, 150, 186

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