Wavelet Coorbit Spaces over Local Fields

TL;DR

This paper extends wavelet coorbit space theory to disconnected local fields, identifying Besov spaces as coorbit spaces and constructing wavelet frames and bases with convergence and unconditionality properties.

Contribution

It applies coorbit space theory to local fields, identifies Besov spaces as coorbit spaces, and constructs explicit wavelet frames and bases with desirable properties.

Findings

Coorbit space theory applies to local fields.

Homogeneous Besov spaces are identified as coorbit spaces.

Explicit tight wavelet frames and orthonormal bases are constructed.

Abstract

This paper studies wavelet coorbit spaces on disconnected local fields $K$, associated to the quasi-regular representation of $G = K \rtimes K^*$ acting on $L^2(K)$. We show that coorbit space theory applies in this context, and identify the homogeneous Besov spaces $\dot{B}_{\alpha,s,t}(K)$ as coorbit spaces. We identify a particularly convenient space $\mathcal{S}_0(K)$ of wavelets that give rise to tight wavelet frames via the action of suitable, easily determined discrete subsets $R \subset G$, and show that the resulting wavelet expansions converge simultaneously in the whole range of coorbit spaces. For orthonormal wavelet bases constructed from elements of $\mathcal{S}_0(K)$, the associated wavelet bases turn out to be unconditional bases for all coorbit spaces. We give explicit constructions of tight wavelet frames and wavelet orthonormal bases to which our results apply.