A Study Of Certain Wavelet Transforms On Hypergroups

Abstract

The present thesis entitled quotA STUDY OF CERTAIN WAVELET TRANSFORMS ON HYPERGROUPSquot is being supplicated for the award of the Doctor of Philosophy in Mathematics. newlineThe thesis consists of seven chapters: newlineThis thesis deals with applied mathematics with wavelets as a joint subject. The theory and newlineapplications of wavelets have undoubtedly dominated the journals in all mathematical, engineering and related fields throughout the last decade. The word quotwaveletquot was first used by newlineMorlet and Grossman in the early 1980s. Several wavelet transform types and hypergroups newlinetypes are examined in this thesis. We also acquired the formulas for inversion and convolution newlineof these wavelets. There have also been discussions about some wavelet transform applications. newlineChapter I: The definition and fundamental characteristics of the Fourier transform have been newlinecovered in this chapter. The wavelet and wavelet transform construction benefits greatly from newlinethese features. Furthermore, the definitions and Elementary properties of the Jacobi polynomial, the Hermite polynomial are provided. These polynomials are crucial to the development newlineof wavelets and wavelet transforms. Additionally, we explain Fourier-Jacobi transform, Hankel transform, Radon transform, the Bessel wavelet transform and Hermite transforms. Finally, newlineGeneralized Harmonic analysis and hypergroups were introduced. newlineChapter II: In this chapter, the definition of Radon transform takes Hermite hypergroup newlineinto consideration. We consider a space made up of infinitely differentiable, rapidly decreasing functions and their derivatives. In addition, we determine an inversion formula and a newlinePlancherel theorem. Through the use of the continuous wavelet transform on the Hermite newlinehypergroup, we get an additional formula for the inverse. newline