Wavelets and related Functions on Cantor Dyadic Group and Vilenkin group

Abstract

The purpose of our work is to study properties of orthonormal wavelets, and the related concepts of frame wavelets and Riesz wavelets, for both Cantor dyadic group and Vilenkin group. At first, we have given the theory of wavelet sets. We have characterized wavelet sets for Cantor dyadic group. All the wavelets originating from wavelet sets are not necessarily associated with a multiresolution analysis (MRA). We have also established relation between wavelets obtained from MRA and wavelets determined by wavelet sets. Scaling and generalized scaling sets provide wavelet sets and hence wavelets. We have given characterization of scaling sets and its generalized version along with relevant examples for Cantor dyadic group. Further, we have studied properties of generalized scaling sets, and relation between wavelet sets and generalized scaling sets. Results related to these sets are also given for the Vilenkin group. Frame multiresolution analysis (FMRA) is an extension of the concept of MRA, and can be used to generate frames. By using properties of shift invariant spaces, relation between FMRA and MRA is established. Further, in a particular case, conditions are given for the existence of frame wavelets associated with FMRA. For Cantor dyadic group generalizations of lowpass filters and scaling functions are introduced, and existence of generalized Parseval frame wavelets is proved. At the end, association between MRA and Riesz wavelets is given along with the construction of frame and Riesz multiwavelets.