Perturbation Of Frames And Biorthogonal M-band Multiresolution Analysis

This paper mainly contain four parts.The first part is introduction .In this part ,we introduce the origin of wavclet,progress and application,otherwise introduce the background and main content of this paper.In the first chapter ,we firstly introduce the definations of frame perturbation and stability of frame perturbation,then we give some conclusions presented by prchumans. Among them Paley —Wiener theorem is the most important, because many conclusions arc obtained from it .Another in this chapter,we give another method that can decide the stability of frame perturbation . That is we resort to the eigenvalue of a special matrix to characterize sufficient condition of stability of frame perturbation.In the second chapter ,we generalize the conclusion presented by Z.Kuang and M.Cui from 2-band to M—band and prove that scaling function and wavelet functions corresponding to generalized â…¡ style filter have also good properties.In the third chapter , we generalize the results of A.Cohen and I.Daubechies. That's suppose we give a pair of M—band dual and finite -impulse response filters m₀(?) and m₀(?),we define function sequences and functions with themWhat's the necessary and sufficient conditions such that functions φ(x)and φ(x) belong to L²(R) and in L²(R) sense φ_n(x) converges to φ(x)and φ_n(x) converges to φ(x)? A. Cohen and I. Daubechies gave an answer when M = 2, we generalize this conclusion to the condition that Mis arbitrary integer that's greater than 2. Especially when M = 3, wegive two examples,the first example ,when M = 3 ,we check the dual and finite-impulse response filters mo{z^),Tho(w) obtained from B— spline function satisfy the criterion of theorem 3.5. The second one ,we discuss the range of scaling sequence and dual scaling sequence satisfied the criterion of theorem under different conditions.In the following we display all the main results in this dissertation .In chapter 1 , and Theorem 1.8.1 and Collary 1.8.1 and Collary 1.8.2 are primary .In chapter 2, the main conclusions are Theorem 2.4 and Theorem 2.5 .In chapter 3 , Lemma3.2 ,Theorem 3.3, Theorem 3.4, Theorem 3.8 and two examples are the most important.