Nonuniform Periodic Wavelet Frames on Non-Archimedean Fields

TL;DR

This paper introduces a new framework for nonuniform periodic wavelet frames on non-Archimedean fields, extending wavelet analysis to non-uniform signals using spectral pairs and Fourier techniques.

Contribution

It develops a novel approach for constructing nonuniform periodic wavelet frames on non-Archimedean fields utilizing Fourier transform and the unitary extension principle.

Findings

Established a new concept of nonuniform periodic wavelet frames on non-Archimedean fields.

Provided a method for constructing these frames using spectral pairs and Fourier analysis.

Extended wavelet analysis to non-uniform signals in non-Archimedean settings.

Abstract

In real life application all signals are not obtained from uniform shifts; so there is a natural question regarding analysis and decompositions of these types of signals by a stable mathematical tool. Gabardo and Nashed and Gabardo and Yu filled this gap by the concept of nonuniform multiresolution analysis and nonuniform wavelets based on the theory of spectral pairs for which 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. In this paper, we introduce a notion of nonuniform periodic wavelet frame on non-Archimedean field. Using Fourier transform technique and the unitary extension principle, we propose an approach for the construction of nonuniform periodic wavelet frames on non-Archimedean fields.