Construction Of M-band Symmetric (Bi) Orthogonal Interpolatory Wavelets

When the scale factor 2 is used to define the wavelets, it is well-known thatany compactly supported orthogonal wavelet, which is symmetric or antisym-metric, is some integer translate or possible sign change of the Haar function.Nevertheless, such scaling function and wavelet do exist for the scale factorM > 2. Due to the advantages in theory and applications, M-band waveletshave received considerable attention in recent years. So construction of M-bandwavelets with desired properties, such as orthogonality, symmetry, interpolation,etc, are the hotspots in wavelets research in recently years.This thesis presents our contribution to the construction of M-band waveletswith desired properties by two methods. It is organized as follows:In chapter 1, we mainly introduce the background and significance ofWavelets Analysis, the present condition of the research in Wavelets Analysis,and outline the main research of this paper.In chapter 2, we mainly introduce the basic theories of M-band MRA,wavelet and related notions.In chapter 3, we pay our attention to the case of M-band compactly sup-ported symmetric orthogonal wavelets. We construct a sequence c containingfree peremeters, which can generate a class of compactly supported symmetricorthogonal scaling functions.In chapter 4, we mainly discuss the construction of compactly supportedbiorthogonal wavelets generated by interpolatory refinable functions. We pro-vide an explicit iterative algorithm to construct the dual function of an inter-polatory refinable function with desired regularity when the factor M = 3. Inthe end, several examples are given.