Fractional Biorthogonal wavelets in $L^2(\mathbb R)$

TL;DR

This paper introduces fractional biorthogonal wavelets in $L^2( eal)$, establishing conditions for their basis properties and biorthogonality, extending wavelet theory within the fractional Fourier transform framework.

Contribution

It provides a new framework for fractional biorthogonal wavelets, including basis conditions and biorthogonality characterizations, advancing wavelet theory in the fractional domain.

Findings

Necessary and sufficient conditions for fractional Riesz bases.

Complete characterization of fractional biorthogonality.

Fractional wavelets generate Riesz bases under mild assumptions.

Abstract

The fractional Fourier transform (FrFT), which is a generalization of the Fourier transform, has become the focus of many research papers in recent years because of its applications in electrical engineering and optics. In this paper, we introduce the notion of fractional biorthogonal wavelets on $\mathbb{R}$ and obtain the necessary and sufficient conditions for the translates of a single function to form the fractional Riesz bases for their closed linear span. We also provide a complete characterization for the fractional biorthogonality of the translates of fractional scaling functions of two fractional MRAs and the associated fractional biorthogonal wavelet families. Moreover, under mild assumptions on the fractional scaling functions and the corresponding fractional wavelets, we show that the fractional wavelets can generate Reisz bases for $L^2(\mathbb R).$.