Construction of Nonuniform Wavelet Frames on Non-Archimedean Fields

TL;DR

This paper develops methods to construct nonuniform wavelet frames over non-Archimedean local fields of positive characteristic, extending previous algorithms to a new mathematical setting with potential applications.

Contribution

It introduces oblique and unitary extension principles for nonuniform wavelet frames on non-Archimedean fields, expanding the theoretical framework beyond real numbers.

Findings

Constructed nonuniform wavelet frames over non-Archimedean fields.

Developed oblique and unitary extension principles for these frames.

Presented an example and discussed potential applications.

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 $\Lambda =\left\{ 0,r/N\right\}+2\,\mathbb Z$ 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 objective of this paper is to develop oblique and unitary extension principles for the construction nonuniform wavelet frames over non-Archimedean Local fields of positive characteristic. An example and some potential applications are also presented.