The Structure Of One Kind Of Special Wavelets

With the deeper study of the theory of wavelets and their more extensive applications, many advantages have emerged gradually. MRA (Multiresolution analysis), as a kind of graceful mathematics thought, is accepted by more and more scientists and engineers. However, accompanying with the application to different fields, there also comes many problems, one of which is asking for more wavelets with different characters.The objective of this paper is to establish a complete characterization of tight frames, and particularly of orthonormal wavelets, for an arbitrary dilation factor a >1, that are generated by a family of finitely many functions in L~2 : = L~2(R) This is a generalization of the fundamental work of Weiss and his colleagues who considered only integer dilations. As an application, we give an example of tight frames generated by one single function for an arbitrary dilation a >1 that possess"good"time-frequency localization. As another application, we also show that there does not exist an orthonormal wavelet with good time-frequency localization when the dilation factor a >1 is irrational such that a jremains irrational for any positive integer j . This answers a question in Daubechies'Ten Lectures book for almost all irrational dilation factors. Other applications include a generalization of the notion of s-elementary wavelets of Dai and Larson to s-elementary wavelet families with arbitrary dilation factors a >1.The"classical"MRA wavelets are probably the most important class of orthonormal wavelets. Many of the better known examples as well as those often used in applications belong to this class. Thus, it was a natural question to find necessary and sufficient conditions for an orthonormal wavelet to be an MRA wavelet. An interesting approach to this involves the dimension function. Here, we prove that the dimension functions of the tight frame wavelets, for an integer dilation factor, are integer valued.