Construction of nonuniform periodic wavelet frames on non-Archimedean fields

Owais Ahmad

doi:10.17951/a.2020.74.2.1-17

Abstract

In real life applications not all signals are obtained by 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 \(\Lambda= \{0,r/N\}+2\mathbb{Z}\) 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 the Fourier transform technique and the unitary extension principle, we propose an approach for the construction of nonuniform periodic wavelet frames on non-Archimedean fields.

Highlights

The notion of frames was first introduced by Duffin and Shaeffer [10] in connection with some deep problems in nonharmonic Fourier series and more with the question of determining when a family of exponentials eiαnt : n ∈ Z is complete for L2[a, b]
Gabardo and Nashed [11], and Gabardo and Yu [12] introduced the notion of nonuniform multiresolution analysis and nonuniform wavelets based on the theory of spectral pairs for which the associated translation set Λ = {0, r/N } + 2 Z is no longer a discrete subgroup of 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 the series of papers [1, 2, 3, 4, 5, 19, 20, 21, 22], we have obtained various results related to wavelet and Gabor frames on non-Archimedean local fields

Summary

Introduction

The notion of frames was first introduced by Duffin and Shaeffer [10] in connection with some deep problems in nonharmonic Fourier series and more with the question of determining when a family of exponentials eiαnt : n ∈ Z is complete for L2[a, b]. Nonuniform frame, wavelet mask, scaling function, Fourier transform. They gave sufficient conditions for constructing tight and dual wavelet frames for any given refinable function φ(x) which generates a multiresolution analysis. Zhang and Saito [26] have constructed general periodic wavelet frames for L2[0, 1] using extension principles.

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Conclusion