Contemporary Mathematics
Volume 464, 2008
Composite Wavelet Transforms: Applications and
Perspectives
Ilham A. Aliev, Boris Rubin, Sinem Sezer, and Simten B. Uyhan
ABSTRACT. We introduce a new concept of the so-called composite wavelet
transforms. These transforms are generated by two components, namely, a
kernel function and a wavelet function (or a measure). The composite wavelet
transforms and the relevant Calder6n-type reproducing formulas constitute a
unified approach to explicit inversion of the Riesz, Bessel, Flett, parabolic and
some other operators of the potential type generated by ordinary (Euclidean)
and generalized (Bessel) translations. This approach is exhibited in the paper.
Another concern is application of the composite wavelet transforms to explicit
inversion of the k-plane Radon transform on lRn. We also discuss in detail a
series of open problems arising in wavelet analysis of Lp-functions of matrix
argument.
Contents
1.
Introduction.
2. Composite wavelet transforms for dilated kernels.
3. Wavelet transforms associated to one-parametric semigroups and inversion
of potentials.
4. Wavelet transforms with the generalized translation operator.
5. Beta-semigroups.
6. Parabolic wavelet transforms.
7. Some applications to inversion of the k-plane Radon transform.
8. Higher-rank composite wavelet transforms and open problems.
References.
1.
Introduction
Continuous wavelet transforms
Wf(x,t)
=Cn
ln
f(y)w
(x~y)
dy,
X
E
!Rn,
t
0,
2000 Mathematics Subject Classification. 42C40, 44A12, 47G10.
Key words and phrases. Wavelet transforms, potentials, semigroups, generalized translation,
Radon transforms, inversion formulas, matrix spaces.
The research was supported by the Scientific Research Project Administration Unit of the
Akdeniz University (Thrkey) and TUBITAK (Thrkey). The second author was also supported by
the NSF grants EPS-0346411 (Louisiana Board of Regents) and DMS-0556157.
@2008 American Mathematical Society
1
http://dx.doi.org/10.1090/conm/464/09074
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