Accession Number : ADA305184

Title :   Affine Systems in L2(Rd): The Analysis of the Analysis Operator.

Descriptive Note : Technical summary rept.,

Corporate Author : WISCONSIN UNIV-MADISON CENTER FOR MATHEMATICAL SCIENCES

Personal Author(s) : Ron, Amos ; Shen, Zuowei

PDF Url : ADA305184

Report Date : DEC 1995

Pagination or Media Count : 35

Abstract : Discrete affine systems are obtained by applying dilations to a given shift-invariant system. The complicated structure of the affine system is due, first and foremost, to the fact that it is not invariant under shifts. Affine frames carry the additional difficulty that they are 'global' in nature: it is the entire interaction between the various dilation levels that determines whether the system is a frame, and not the behaviour of the system within one dilation level. We completely unravel the structure of the affine system with the aid of two new notions: the affine product, and a quasi-affine system. This leads to a characterization of affine frames; the induced characterization of tight affine frames is in terms of exact orthogonality relations that the wavelets should satisfy on the Fourier domain. Several results, such as a general oversampling theorem follow from these characterizations. Most importantly, the affine product can be factored during a multiresolution analysis construction, and this leads to a complete characterization of all tight frames that can be constructed by such methods. Moreover, this characterization suggests very simple sufficient conditions for constructing tight frames from multiresolution. Of particular importance are the facts that the underlying scaling function does not need to satisfy any a priori conditions, and that the freedom offered by redundancy can be fully exploited in these constructions. (AN)

Descriptors :   *MATRICES(MATHEMATICS), *SYSTEMS ANALYSIS, *OPERATORS(MATHEMATICS), RESOLUTION, SHIFTING, SCALING FACTOR, INVARIANCE, MAPPING(TRANSFORMATIONS), ORTHOGONALITY, FOURIER ANALYSIS, PERIODIC FUNCTIONS, BESSEL FUNCTIONS.

Subject Categories : Numerical Mathematics
      Operations Research

Distribution Statement : APPROVED FOR PUBLIC RELEASE