Unitary matrix functions, wavelet algorithms, and structural properties of wavelets

Some connections between operator theory and wavelet analysis: Since the mid eighties, it has become clear that key tools in wavelet analysis rely crucially on operator theory. While isolated variations of wavelets, and wavelet constructions had previously been known, since Haar in 1910, it was the advent of multiresolutions, and subband filtering techniques which provided the tools for our ability to now easily create efficient algorithms, ready for a rich variety of applications to practical tasks. Part of the underpinning for this development in wavelet analysis is operator theory. This will be presented in the lectures, and we will also point to a number of developments in operator theory which in turn derive from wavelet problems, but which are of independent interest in mathematics. Some of the material will build on chapters in a new wavelet book, co-authored by the speaker and Ola Bratteli, see http://www.math.uiowa.edu/~jorgen/ .Comment: 63 pages, 10 figures/tables, LaTeX2e ("mrv9x6" document class), Contribution by Palle E. T. Jorgensen to the Tutorial Sessions, Program: ``Functional and harmonic analyses of wavelets and frames,'' 4-7 August 2004, Organizers: Judith Packer, Qiyu Sun, Wai Shing Tang. v2 adds Section 2.3.4, "Matrix completion" with reference

Computing the Sobolev Regularity of Refinable Functions by the Arnoldi Method

The recent paper [J. Approx. Theory, 106 (2000), pp. 185--225] provides a complete characterization of the L 2 -smoothness of a refinable function in terms of the spectrum of an associated operator. Based on this theory, we devise in this paper a numerically stable algorithm for calculating that smoothness parameter, employing the deflated Arnoldi method to this end. The algorithm is coded in Matlab, and details of the numerical implementation are discussed, together with some of the numerical experiments. The algorithm is designed to handle large masks, as well as masks of refinable functions with unstable shifts. This latter case is particularly important, in view of the recent developments in the area of wavelet frames. Key words. refinable functions, wavelets, smoothness, regularity, transition operators, transfer operators, Arnoldi's method AMS subject classifications. Primary, 42C15; Secondary, 39B99, 46E35 PII. S0895479899363010 1

Computing the Sobolev regularity of refinable functions by the Arnoldi Method

The recent paper [RS1] provides a complete characterization of the L2-smoothness of a refinable function in terms of the spectrum of an associated operator. Based on this theory, we devise in this paper a numerically stable algorithm for calculating that smoothness parameter, employing the deflated Arnoldi method to this end. The algorithm is coded in Matlab, and details of the numerical implementation are discussed, together with some of the numerical experiments. The algorithm is designed to handle large masks, as well as masks of refinable function with unstable shifts. This latter case is particularly important, in view of the recent developments in the area of wavelet frames

Computing the Sobolev regularity of refinable functions by the Arnoldi Method

The recent paper [RS1] provides a complete characterization of the L 2 -smoothness of a refinable function in terms of the spectrum of an associated operator. Based on this theory, we devise in this paper a numerically stable algorithm for calculating that smoothness parameter, employing the deated Arnoldi method to this end. The algorithm is coded in Matlab, and details of the numerical implementation are discussed, together with some of the numerical experiments. The algorithm is designed to handle large masks, as well as masks of refinable function with unstable shifts. This latter case is particularly important, in view of the recent developments in the area of wavelet frames