Nonuniform low-pass filters on non Archimedean local fields

Abstract

In real life application all signals are not obtained from uniform shifts; so there is a natural question regarding analysis and decompositions of this types of signals by a stable mathematical tool. Gabardo and Nashed (J. Funct. Anal. 158:209-241, 1998) filled this gap by the concept of nonuniform multiresolution analysis. In this setting, the associated translation set \(\Lambda =\left\{ 0,r/N\right\}+2\,\mathbb Z\) is no longer a discrete subgroup of \(\mathbb R\) but a spectrum associated with a certain one-dimensional spectral pair and the associated dilation is an even positive integer related to the given spectral pair. The main aim of this article is to provide the characterization of nonuniform low-pass filters on non-Archimedean local fields.