Introduction to Partial Differential Equations with MATLAB

(4.59)

0, 2. 4

—d. Then

(R°'f)(x) =

(4.60)

Proof. By Theorem I and Lemma 4. / is well defined, and almost all x. Hence (4.60) holds in view of Corollary 3.

Remark. If I a > 0, a + d —

is finite for

then, by Corollary 3. the equality (4.60) holds for all The limit case a = 0 gives (4.53). Furthermore, by (4.52), K°f can be regarded as the identity operator. Thus (4.60) means that the n

0. 2.4

inverse off is just

the analytic continuation of (4.59) at the point a = —d (at least formally). Various applications of the remarkable equality (4.60) are given in lRlOJ.

5 The convolution-backprojection method and wavelet-like transforms The general scheme of the convolution-backprojection method in the hyperbolic setup is as follows. Given a measurable function w(s) on (0, let

=

XE X.

(5.61)

Owing to G-invariance, we have

(Wf)(x) =

f

By (3.37). a formal application of (2.31) yields

(5.62)

C.A. Berenstein and B. Rubin

52

(Wf)(x)

— 1)dy,

=

=

c

=

(5.63)

where

j(t

= is

—

the Riemann-.Liouville fractional integral. Thus

(Wf)(x)

=

f

y]2

—

(5.64)

1)dy.

lf fx f(y)ifr([x. yJ2 — l)dy is an approximate identity on X, then (5.64) enables us to recover f from!. Now we discuss details.

Lemma 5. Let! E w(s)

p < (n — 1)/(d

I

ifs

1.

o(s_n_1/P'—E2) ifs

I.

I

=

for sufficiently small

— 1).

Suppose that

I/p + lip' = 1.

(5.65)

e2 > 0. Then (5.64) holds for almost all x.

Proof. We split (5.61) into two integrals over the sets

> I}.

< 1),

By (5.65), WI I

C

(Ii + 12) where c = c(a) = const,

i= 1.2;

I =

l—d—e2+(n—I)/p.

flu

By Theorem 2. is finite a.e. Thus the integral (5.62) with f replaced by Ill is finite a.e., and therefore (2.31) is applicable. This gives (5.64). I

We denote

(W,ço)(x) =

!

: >0,

(5.66)

where the function w will be at our disposal. This operator is a scaled version of (5.61). Suppose that w(s) obeys (5.65). By Lemma 5,

(W,f)(x)

=

f

fO')*(Ex. y]2 — l)dy.

=

CT I—n/2

A(.),

(5.67)

Radon Transform

=

53

(5.68)

c being the constant from (5.63). The right-hand side of the first equality in (5.67) represents an approximate identity of the form

(K6f)(x)

y])f(y)dy,

= Jx

=

s>

koI

C

0.

Theorem 3. Let f be a measurable function on X. We introduce a maximal operator

(K*f)(x) = sup I(K€f)(x)I. The following statements hold.

I. If ko(s) has a decreasing integrable majoran:, then Kf
f*(x) = sup

r>O IB(x,r)l

jf

B(x.r)

X : dist(x, y) = cosh'[x, yJ
B(x, r) = {y

II. 1ff E LP(X), I