2 ILHAM A. ALIEV, BORIS RUBIN, SINEM SEZER, AND SIMTEN B. UYHAN
where
w
is an integrable radial function satisfying
fJRn
w(x)dx
=
0, have proved to
be a powerful tool in analysis and applications. There is a vast literature on this
subject (see, e.g.,
[Da], [HO], [M],
just for few). Owing to the formula
(1.1)
roo
ill - a~
Jo W f(x, t) tHa - Ca,w(
-~)
f(x),
aEC,
that can be given precise meaning, continuous wavelet transforms enable us to
resolve a variety of problems dealing with powers of differential operators. Such
problems arise, e.g., in potential theory, fractional calculus, and integral geometry;
see,
[HO], [Rl]-[R7], [Tr].
Dealing with functions of several variables, it is always
tempting to reduce the dimension of the domain of the wavelet function w and find
new tools to gain extra flexibility. This is actually a motivation for our article.
We introduce a new concept of the so-called composite wavelet transforms.
Loosely speaking, this is a class of wavelet-like transforms generated by two com-
ponents, namely, a kernel function and a wavelet. Both are in our disposal. The
first one depends on as many variables as we need for our problem. The second
component, which is a wavelet function (or a measure), depends only on one vari-
able. Such transforms are usually associated with one-parametric semigroups, like
Poisson, Gauss-Weierstrass, or metaharmonic ones, and can be implemented to
obtain explicit inversion formulas for diverse operators of the potential type and
fractional integrals. These arise in integral geometry in a canonical way; see, e.g.,
[H, R2, R6, R9].
In the present article we study different types of composite wavelet transforms
in the framework of the Lp-theory and the relevant Fourier and Fourier-Bessel
harmonic analysis. The main focus is reproducing formulas of Calderon's type
and explicit inversion of Riesz, Bessel, Flett, parabolic, and some other potentials.
Apart of a brief review of recent developments in the area, the paper contains
a series of new results. These include wavelet transforms for dilated kernels and
wavelet transforms generated by Beta-semigroups associated to multiplication by
exp(
-tl~li3),
f3
0, in terms of the Fourier transform. Such semigroups arise in the
context of stable random processes in probability and enjoy a number of remarkable
properties
[Ko], [La].
Special emphasis is made on detailed discussion of open
problems arising in wavelet analysis of functions of matrix argument. Important
results for £ 2-functions in this "higher-rank" set-up were obtained in
[OOR]
using
the Fourier transform technique. The
Lp-case
for p
=I
2 is still mysterious. The
main difficulties are related to correct definition and handling of admissible wavelet
functions on the cone of positive definite symmetric matrices.
The paper is organized according to the Contents presented above.
2. Composite Wavelet Transforms for Dilated Kernels
2.1. Preliminaries.
Let
Lp
=
Lp(lR.n),
1
~
p oo, be the standard space of
functions with the norm
II/IlP
=
(1n
lf(xW dx
f
1
p
oo.
For technical reasons, the notation
£
00
will be used for the space
C0
=
C0 (JR.n)
of
all continuous functions on
JR.n
vanishing at infinity. The Fourier transform of a
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