Nonuniform Periodic Wavelet Frames on Non-Archimedean 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 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 $Λ=\left\{ 0,r/N\right\}+2\,\mathbb Z$ 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.