Date of Submission


Date of Award


Institute Name (Publisher)

Indian Statistical Institute

Document Type

Doctoral Thesis

Degree Name

Doctor of Philosophy

Subject Name



Theoretical Statistics and Mathematics Unit (TSMU-Kolkata)


Behera, Biswaranjan (TSMU-Kolkata; ISI)

Abstract (Summary of the Work)

In this chapter, we will give a brief history of wavelet analysis on R. We will also list some bask results on local felds which will be used in subvequent chapters.1.1 Wavelets on RWe fiest start with a brief history of wavelets und some basic defnitions and results conceming the orthonormal wavekts on R.1.1.1 A brief historyIn the last few decades vaveler theory has growa extensively and has drawn great atlention sot only in mathematies bu also in engineering, pitysics, computer science and many other fields. In signal and image processing, wavelets play a very important role.In 1910, A. Haar grve the first example of an orthonormal wavelet oa R but because of the poor frequency localization of the resulting onhonormal basis, they are not of much use in practice. In 1981, while trying to further understand the Hardy spaces, Sumberg (71] obtained a waveles of L(R) by modtying a basis constructed carlier by Franklis ia 1927. We refer to (75) for a detailed discusin of the Strömberg wavelet. In the early eighies, Morlet introduced the cantinuous wavelet vansform. Crossman obtained an inversion formala for this trinsform and along with Morlet explored several applications. Meyer (64] constracled an example of an infinitely differertiable wavelet such tha its Fourier transform also had this property. This construction was generalized to higher dimensions by Lomarié and Meyer (58). The concept of ultiresolution analysis (MRA) waa developed by Meyer and Mallat (63, 65). Duubechies uned this conopt to construct compacty supported wavelets with artitrarily high, tut fixed, regulutty.The wavelets have poor trequency kocalization. To overcome this disadvantapo, Cofnan, Meyer and Wickerhauser (27] constructed wavelet packets from a wavelet associated with an MRA. Colen, Daubechies and Feanvesu in (25] introduced the concept of biorthogonal vanelets. We will discuss these concepts in details in subsequent chapters.Warwekets and multiresolution srelyses were also studiedestensively in the higher dinersiceal cases see (20. 30, 42, 62, 65, 75) and references therein. The concept of wavelet has been Calended to many different setups by several authors. Dahike (291 introduced it on locally compact abelan groups (see also (32. 43). It was generalized o abstract Hilbert spuces by Han. Larson, Papadakis and Stavroposlos (39, 70). Lemarie 56) extended this concept o vraified Lie groups. Recently, R. L. Benedeno and J. J. Benedetno [) developed a wavelet theory for local fields and related groups. In [12). R. L. Benedetto proved that Haae and Shanten vavekets exist and, in fact, both are the same for such a group.1.1.2 Basic concepts of waveletsIn this section we wili discuss some basic delinitions and results which are useful in the theny of waveleta.Definition L.1.1. A collection {s. :e Z) of functions in L(R) is called an orthenomal system if it satisfies(a. ) - Sma m, neZ, where 1, fm=n, 0. f man.


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