COMPACTLY SUPPORTED (BI)ORTHOGONAL WAVELETS GENERATED BY INTERPOLATORY REFINABLE FUNCTIONS

Abstract

Univariate compactly supported fundamental refinable functions and wavelets have been studied extensively. However, for higher dimensions, there are only a few isolated nouseparahle examples with their regularities determined. The main results of this thesis are the constructions of nonseparable fundamental refinable functions and biorthogonal wavelets for any number of variables and for any dilation matrix. We present a general method to construct such functions with arbitrary regularity and symmetry provided an appropriate initial interpolatory refinable function. vVe also provide an asymptotic analysis of regularity for these constructions and review a criterion of smoothness for practical purpose. In the thesis, a general method to construct smooth compactly supported fundamental orthonormal symmetric refinable functions with refinable functions with dilation factor >3 is developed and this leads to the construction of symmetric wavelets.