Biorthogonal Wavelets

Abstract

In this chapter, we introduce the concept of biorthogonal wavelets in a local field K of positive characteristic. We show that if $\varphi$ and $\tilde{\varphi }$ are the scaling functions of two multiresolution analyses (MRAs) such that their translates are biorthogonal, then the associated families of wavelets are also biorthogonal. Under mild decay conditions on the scaling functions and the wavelets, we also show that the wavelets generate Riesz bases for $L^2(K)$. First, we find necessary and sufficient conditions for the translates of a function to form a Riesz basis for their closed linear span. We define the projection operators associated with the MRAs and show that they are uniformly bounded on $L^2(K)$. Finally, we prove that the wavelets associated with dual MRAs are biorthogonal and generate Riesz bases for $L^2(K)$.