Dual wavelets associated with nonuniform MRA

Abstract

A generalization of Mallats classical multiresolution analysis, based on thetheory of spectral pairs, was considered in two articles by Gabardo and Nashed. In thissetting, the associated translation set is no longer a discrete subgroup of R but a spectrumassociated with a certain one-dimensional spectral pair and the associated dilation is aneven positive integer related to the given spectral pair. In this paper, we construct dualwavelets which are associated with Nonuniform Multiresolution Analysis. We show thatif the translates of the scaling functions of two multiresolution analyses are biorthogonal,then the associated wavelet families are also biorthogonal. Under mild assumptions onthe scaling functions and the wavelets, we also show that the wavelets generate Rieszbases

On the nonhomogeneous wavelet bi-frames for reducing subspaces of Hs(K)

“…Although their topology is similar, their MRA (multiresolution analysis) and wavelet theories are quite different. The construction of wavelets, wavelet frames, MRA and other related works on local fields of positive characteristics (LFPC)have been studied by Benedetto, Behera and Jahan, Ahmad and Shah, Jiang, Li and Ji, Shukla and Mittal, Bhat, etc in the series of works [2,3,4,5,6,7,8,9,10,11,12,13,14,15,17,25,26,27]. However the research on local fields is still in its infancy.…”