Nonuniform Discrete Wavelets on Local Fields of Positive Characteristic

Abstract

A constructive algorithm based on the theory of spectral pairs for constructing nonuniform wavelet basis in $L^2(\mathbb R)$ was considered by Gabardo and Nashed (J Funct Anal 158:209–241, 1998). In this setting, the associated translation set is a spectrum $\Lambda$ which is not necessarily a group nor a uniform discrete set, given $\Lambda =\left\{ 0,r/N\right\} +2\,{\mathbb {Z}},$ where $N\ge 1$ (an integer) and r is an odd integer with $1 \le r \le 2N -1$ such that r and N are relatively prime and $\mathbb Z$ is the set of all integers. The objective of this paper is to develop nonuniform discrete wavelets on local fields. We first provide a characterization of an orthonormal basis for the Hilbert space $l^2(\lambda )$ and then show that it can be expressed as orthogonal decomposition in terms of countable number of its closed subspaces. Moreover, we show that the wavelets associated with NUMRA on local fields of positive characteristic are connected with the wavelets on spectrum.