A new look at compactly supported biorthogonal Riesz bases of wavelets

Abstract

In this paper, we give two algorithms to compute compactly supported biorthogonal Riesz basis of wavelets. The input to these algorithms are filters of the transfer and the dual transfer functions, which are obtained by solving the Bezout equation. This Bezout equation arises from biorthogonality of the scaling function and the dual scaling function. We solve the Bezout equation in a simple and algebraic way. We also give a case study of the biorthogonal wavelets showing their detail construction. Some references to Sobolev regularity of the wavelets which is a qualitative property of the wavelet is also made by us.