On the vaguelet and Riesz properties of L^2-unbounded transformations of orthogonal wavelet bases

TL;DR

This paper investigates conditions under which certain unbounded transformations of orthogonal wavelet bases produce vaguelets and Riesz bases, with implications for wavelet-based stochastic process decompositions.

Contribution

It establishes criteria for when these transformations generate vaguelets and Riesz bases, including examples and counterexamples based on quasi-homogeneity.

Findings

Transformations with quasi-homogeneous functions produce vaguelets.

Some non-quasi-homogeneous functions do not generate vaguelets.

The Riesz basis property can be inferred from vaguelet properties.

Abstract

In this work, we prove that certain L^2-unbounded transformations of orthogonal wavelet bases generate vaguelets. The L^2-unbounded functions involved in the transformations are assumed to be quasi-homogeneous at high frequencies. We provide natural examples of functions which are not quasi-homogeneous and for which the resulting transformations are not vaguelets. We also address the related question of whether the considered family of functions is a Riesz basis in L^2(R). The Riesz property could be deduced directly from the results available in the literature or, as we outline, by using the vaguelet property in the context of this work. The considered families of functions arise in wavelet-based decompositions of stochastic processes with uncorrelated coefficients.