Filter Frequency Response And Wavelet Sets

The progenitor of wavelet analysis is Fourier analysis, which has been used broadly but, the same time, has its shortcomings that is not negligible. That is, it cannot well describe the locality of functions, which greatly restricts its applications. As a fairly new discipline developed gradually on the base of Fourier analysis, not only can wavelet analysis share the advantages of the former, but also can make up its deficiencies, and because of which, it has been growing rapidly since the day of its birth. One of the most essential questions of wavelet analysis is the construction of wavelet, and multiresolution analysis is the important and central method to carry it out. Therefore, the relation of them becomes a significant topic of wavelet analysis. R.A.Zalik gave the one dimension case of the relation of Riesz wavelet and wavelet in [35], which includes a sufficient and necessary condition, some equivalent conditions and concrete examples. Beginning from the conclusion in this paper, I have got several new results on Riesz wavelet and MRA.This paper is composed of three sections.The first section includes two parts.The first part gives out the definitions of frame, Riesz base, multiresolution analysis (MRA) and its relation to wavelet, thus generalizes a relation between Riesz wavelet and MRAto high dimension, which was stated in a theorem in Zalik [35]. Then, we apply the sufficient-necessary condition in [35], judging whether a Riesz wavelet is come from an MRA, to some concrete examples, and change a condition in this theorem into a more intu-itionistic one. At last there is a theorem on the range of the space generated by dilations and tranlations of Riesz wavelets.For the second part, a property of filter is first stated: a matrix with respect to lowpass and highpass filter is of full rank, when the Riesz wavelet is obtained from an MRA; the second theorem proves a property of frequency response of high-dimensional MRA.In the second section, the author depicts the frames and Riesz bases on Euclidean spaces. First, a sufficient-necessary condition on which m vectors (m > d) on Rd is tight frame on Rd is demonstrated; second, a sufficient condition for general frames and Riesz bases; third, a concrete instance of the frame on Rd; finally, a property concerning the matrix made up of vectors that form a frame.The final section is a general report about wavelet sets summarizing the latest results of recent years. The author begins with the simplest case: 2-adic dilation, 1-dimensional integral translation; and then generalizes step by step: first from 2-adic dilation to A-adic dilation, second the translation operator to d-dimansional integer space Zd. and further some results related to wavelet sets on subspaces of L2, and MRA.