A Study On Certain Type Of Wavelet Packets And Wavelet Transforms

Abstract

The present thesis entitled quotA Study on Certain Type of Wavelet Packets and Wavelet newline Transformsquot is the outcome of the investigations carried out by the author towards the newline fulfillment for the award of the degree of Doctor of Philosophy. newline In 1980, Morlet and Grossmann developed the theory of wavelets and wavelet transforms. newline This concept is being developed daily by many authors. In this thesis, we discussed vari newlineous types of wavelet transforms, wavelet packets, and wave packet transforms associated newline with different functions, including integral transforms. This thesis consists of eight chap newlineters, beginning with an introduction in chapter 1. newline Chapter I: This chapter provides an overview of the Fourier transform, wavelet trans newlineform, multiresolution analysis, local fields, locally compact abelian groups, index Whit newlinetaker transform, and various other functions used in the study. Finally, the chapter newline concludes with a literature review. newline Chapter II: In this chapter, a reconstruction and inversion formula for the continuous newline wavelet transform on abelian group for band-limited functions is defined. This formula newline possesses a more explicit expression than the well-known result. Also, Parseval and other newline interesting results on abelian groups are obtained. newline Chapter III: The objective of this chapter is to construct wavelet packets associated newline with multiresolution analysis on local fields. Moreover, from the collection of dilations newline and translations of the wavelet packets, we characterise the subcollections which form an newline orthonormal basis for L2(K). newline Chapter IV: In this chapter, using the harmonic analysis associated with the Bessel newline function on [0,+and#8734;[, we investigate two categories of generalised wavelet packets and the newline corresponding generalised wavelet transforms, as well as the Plancherel, Calderon, and newline reconstruction formulas for these transforms. newline Chapter V: In this chapter, we construct a wavelet frame system on local field K newline associated with the multiresolution analysis and Haar measures.