Multiresolutions, multivariable operator theory, and noncommutative analysis
MAM intensive research course for PhD students and researchers.
MAM research frontier lecture series.
Lecture 2: Analysis, representation theory, and filters
Multiresolutions, multivariable operator theory, and Noncommutative Analysis
We offer an operator theoretic approach to multiresolutions in noncommutative analysis and applications. Multiresolutions are part of multivariable operator theory, and noncommutative mathematics. In applications, they are important as they offer fast algorithms, and have a host of other applications. One such is to wavelets; a proven and successful alternative to classical Fourier methods, Fourier series and integrals. In general, with multiresolutions, one obtains recursive and computational spectral resolutions for multivariable operator systems. They are localized, so better adapted to discontinuities. And they offer better numerical schemes.
Multiresolutions are further useful in the study of self-similarity, in the analysis of fractals, and of non-linear dynamical systems. A special case of this is illustrated by the renormalization property for scaling functions from wavelet theory; and renormalization more generally. Recursive multiresolutions basis constructions in Hilbert spaces are key tools in analysis of fractals and of iterated function systems in dynamics. Indeed, with multiresolutions, notions of selfsimilarity and locality yield much better pointwise approximations than is possible for traditional Fourier bases.
This lecture will further cover frequency sub-bands in signal or image-processing, and associated multi-band filters: One builds recursive subdivisions of signals into frequency bands. This suggests a representation theoretic framework; realizations of certain representations on Hilbert spaces H having associated families of closed subspaces in such a way that "non-overlapping frequency bands" correspond to orthogonal subspaces in H; or equivalently to systems of orthogonal projections. Since the different frequency bands must exhaust the range for signals in the entire system, one looks for orthogonal projections which add to the identity operator in H. Representations of Cuntz algebras is a case in point. From representations we obtain classification of families of multi-band filters; and representations allow us to deal with non-commutativity as it appears in both time/frequency analysis, and in scale-similarity. The representations further offer canonical selections of special families of commuting orthogonal projections.
Analysis, representation theory, and filters.
We study subdivision of analogue-signals into frequency bands in signal/image-processing. Motivated by applications to digital filters we suggest a new representation theoretic framework. We build particular representations creating both Hilbert space H and algebra representing digital subdivisions. This leads to a filtered system of closed subspaces in H such that "non-overlapping frequency bands" correspond to orthogonal subspaces in H; or equivalently to systems of orthogonal projections.
Since the different frequency bands must exhaust the signals for the entire system, one looks for orthogonal projections which add to the identity operator in H. Since time/frequency analysis is non-commutative, one is further faced with a selection of special families of commuting orthogonal projections. From this and repeated subdivision sequences we generate recursive algorithms for new bases and frames including wavelet families.