Orthogonal Multiwavelet And Multiresolution Analysis

Abstract. Wavelet analysis is an applicated department began in 80th years of 20 century, which is an improvement and develepment of traditional Fourier analysis and is widely used in areas, such as compression figure, signal processing ,fractal and so on. Multiwavelets is an newly department recently, since it has some advantage different from wavelet. Based on some existing papers, we mainly characterize orthogonal multiwavelets and its construction, equivalent conditions of Riesz multiwavelet associated with multiresolution analysis, characterisation of a pair of multiwavelets assotiated with multiresolution analysis. This thesis consists of four chapters. The main results are as follows.In chapter 1, we concentrate on the preliminaries including some notations and definitions, such as Bessel sequence, frame, Riesz basis, orthogonal wavelet and Riesz wavelet and so on, then give some well-known theorems.In chapter 2, first of all,we characterize subspace V₀ of multiresultion analysis {V_j}_-∞^+∞ based on invariant subspace. Subsequently, we consider the equivalent conditions among orthogonal multiwavelet Ψ = (Ψ₁,Ψ₂,...Ψ_r)^T, subspace, basis and dimension of subspace, properties of filter function matrix P(w) are dealt with. Lastly, we give some conditions of an function to be an scaling function.In chapter 3, having studied relation between filter function matrices associated with Fourier transform of scaling function Φ = (Φ₁,Φ₂,...,Φ_r)^T and multiwavelet Ψ = (Ψ₁,Ψ₂,...Ψ_r)^T, we discuss relationship on basis of subspaces V₀, V₁ and filter function matrix; In addition, we prove the existence of orthogonal multiwavelet and construct it.In chapter 4, we convert to the properties of Riesz multiwavelet,and bring out a set of equivalent conditions among Riesz multiwavelets Ψ, scaling function Φ and low, high filter function matrix; Having analylized the conditions of two function transformed of multiwavelet associated with multisolution analysis,dimension of function matrix M(w) and multiwavelet Ψ. Finally, we prove the main result of a pair of biorthogonal multiwavelets associated with multisolution analysis.