Multiwavelet Sets, Multiframelet Sets and Related Systems on Qp

Abstract

The work done in this dissertation revolves around wavelets, MRAs and frames on the p-adic field Qp. It is well known that every element of a separable Hilbert space can be represented in terms of its orthonormal basis by using the respective Fourier coefficients. Multiwavelets serve this purpose in wavelet theory. Multiwavelet is a collection of finitely many functions whose dilates followed by translates give an orthonormal basis. Multiframelets are generalization of multiwavelets. Multiframelets provide more flexibility in the representation of an element of the space under consideration. These concepts were initially studied for the Euclidean space Rn (n ≥ 1) and later on this study was extended to more general spaces like local fields of positive characteristic. In the same spirit, here we have established results related to muliwavelets/multiframelets for Qp. Multiwavelet sets/multiframelet sets provide a special class of muliwavelets/multiframelets. Multiwavelet and multiframelet sets can be constructed by using scaling sets, generalized scaling sets and frame scaling sets. These sets are not studied in Qp. MRAs also generate wavelets and a generalization of MRA is FMRA. Results related to MRA in L2(Qp) are available in literature but as per our knowledge FMRAs in L2(Qp) are not studied. The dissertation is divided into nine chapters. First two chapters contain basic definitions and results. Next six chapters are devoted to the study of above mentioned concepts in L2(Qp) and contain their properties, some characterizations and related examples. Finally, some problems for future work are given in Chapter 9.