Special Affine Biorthogonal Wavelets in $L^2(\mathbb R)$ and on a Logarithmic Regression Curve

Abstract

In the article “Special affine multiresolution analysis and the construction of orthonormal
wavelets in L2(R)”, [Appl Anal. 2022; D.O.I: 10.1080/00036811.2022.2030723], we introduced the notion
of multiresolution analysis (MRA) in the realm of the special affine Fourier transform. As a continuation
of the study, we present the construction of special affine biorthogonal wavelets in L2(R) and establish
a complete characterization for such wavelets in terms of the biorthogonality of translates of the scaling
functions of two special affine MRA’s. To extend the scope of the present study, we introduce the notion of
special affine biorthogonal MRA on a logarithmic regression curve C and investigate all its fundamental
properties associated with a real, augmented matrix M = (A, B,C,D : p, q).