Non-MSF Mra Wavelets

Abstract

In this article, we provide two classes of non-MSF MRA wavelets in L 2 (R 2 ).The first arose through one-dimensional dyadic wavelet sets having two components, is an uncountable family, while the second one arose through a special kind of MRA wavelet sets having three components, is a countable family.

Minimally Supported Frequency (MSF) d-Dilation Wavelets

“…Wavelet sets were first introduced by Dai and Larson [4]. Wavelet sets having certain number of components were studied by many researchers in [2,3,[5][6][7][8][9][10][11][12]. Characterizations of two-, three-, and four-interval wavelet sets were obtained by Ha et al for dilation 2 in [2].…”

“…Wavelets and wavelet sets were extensively studied, considering various aspects such as the multiresolution analysis and the path connectivity of wavelets, by several researchers in the field of wavelets [10][11][12]14,15]. From a result by Chui and Shi [1], according to which for a dilation d such that d j / ∈ Q for all j ∈ N, the only wavelets that exist are MSF wavelets, the question of the existence or otherwise of a non-MSF wavelet for dilation other than the one considered by them arose.…”