PERIODIC WAVELETS

Abstract

Wavelets are recent development in mathematics with wide applications. They have already been useful in signal analysis, image processing and numerical analysis. This wide applicability is one of the factors that generates so much interest in the subject in recent years.

In this report, we shall study wavelets in L2 [0, 2 ?), the space of 2 ?-periodic, square-integrable functions. Unlike the non.-periodic space L2 (R ), the wavelets may not be generated from a single function.

A definition of a multiresolution analysis of L2 [O, 2 ?) is given in Chapter 2 where necessary and sufficient conditions for the characterization of scaling functions are derived. The rnultiresolution analysis defined here is more general than one obtained by periodizing a multiresolution analysis of L 2 ( R ), and is a periodic analogue of the non-stationary multiresolution introduced by de Boor, DeVore and Ron [2]. A periodic analogue of a theorem of de Boor, DeVore and Ron on the denseness of the union of {Vk}k ? z in L2(R ) is given in Section 3. The main feature in the periodic case is the availability of the orthogonal splines which are used in the construction of wavelets. The constructions of scaling functions and wavelets are done in Sections 4 and 5 respectively. Another way of constructing wavelets is by considering the dual basis. This is clone in Section 6. The orthogonal splines provide an alternative to wavelets in the decomposition and reconstruction of periodic square-integrable functions. The orthogonal spline algorithms are studied in Section 7.

In the last chapter, the general theory is applied to the construction of periodic versions of the compactly supported wavelets of Daubechies, the non­orthogonal cardinal spline wavelets of Chui and Wang mid the trigonometric wavelets of Chui and Mhaskar.