Necessary condition and sufficient conditions for nonuniform wavelet frames in L2(K)

Abstract

A constructive algorithm based on the theory of spectral pairs for constructing nonuniform wavelet basis in [Formula: see text] was considered by Gabardo and Nashed [Nonuniform multiresolution analysis and spectral pairs, J. Funct. Anal. 158 (1998) 209–241]. In this setting, the associated translation set is a spectrum [Formula: see text] which is not necessarily a group nor a uniform discrete set, given [Formula: see text] where [Formula: see text] (an integer) and [Formula: see text] is an odd integer with [Formula: see text] such that [Formula: see text] and [Formula: see text] are relatively prime and [Formula: see text] is the set of all integers. The objective of this paper is to construct nonuniform wavelet frame on local fields. A necessary condition and four sufficient conditions for nonuniform wavelet frame on local fields are given.