Nonuniform Multiresolution Analyses and Spectral Pairs

✍ Scribed by Jean-Pierre Gabardo; M.Zuhair Nashed

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 421 KB
• Volume: 158
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1998.3253

✦ Synopsis

A generalization of the notion of multiresolution analysis, based on the theory of spectral pairs, is considered. In contrast to the standard setting, the associated subspace V 0 of L 2 (R) has, as an orthonormal basis, a collection of translates of the scaling function , of the form [,(x&*)] * # 4 where 4=[0, rÂN]+2Z, N 1 is an integer, and r is an odd integer with 1 r 2N&1 such that r and N are relatively prime and Z is the set of all integers. Furthermore, the corresponding dilation factor is 2N, the case where N=1 corresponding to the usual definition of a multiresolution analysis with dilation factor 2. A necessary and sufficient condition for the existence of associated wavelets, which is always satisfied when N=1 or 2, is obtained and is shown to always hold if the Fourier transform of , is a constant multiple of the characteristic function of a set.