Construction of Nonuniform Wavelet Frames on Non-Archimedean Fields

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.