Wavelet Coorbit Spaces over Local Fields

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.