Composite Wavelet Transforms: Applications and Perspectives

TL;DR

This paper introduces composite wavelet transforms combining kernel and wavelet functions, providing a unified framework for inverting various potential-type operators and the Radon transform, with discussions on open problems in wavelet analysis of matrix arguments.

Contribution

It presents a novel composite wavelet transform framework for explicit inversion of potential operators and the Radon transform, unifying several classical approaches.

Findings

Unified approach to inverting potential operators

Explicit inversion of the Radon transform using composite wavelets

Discussion of open problems in wavelet analysis of matrix functions

Abstract

We introduce a new concept of the so-called {\it 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 Calder\'{o}n-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 $\bbr^n$. We also discuss in detail a series of open problems arising in wavelet analysis of $L_p$-functions of matrix argument.