Some Problems in Orthogonal Polynomials and Wavelets

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Proposal: DMS-9970613Principal Investigator: Jeffrey S. GeronimoAbstract: Geronimo will study polynomials orthogonal on several subarcs of the unit circle in order to determine the analog in that context of the Toda flow associated with polynomials orthogonal on finite segments of the real line. He will continue to investigate the properties of two-dimensional pseudopolynomials orthogonal on the bi-circle and their connection to two-dimensional extension problems. This will be related to the factorization of trigonometric polynomials in two variables, a topic that has important applications in wavelet theory. Geronimo also plans to use intertwining analysis to investigate the theory and construction of orthogonal wavelet bases in one and higher dimensions. He will use the results obtained to help study orthogonal multiresolution analyses for quasicrystals.Wavelets and orthogonal polynomials have proven themselves to be important tools in attacking complex, real-world problems such as the analysis and compression of images and signals (like those found on the internet), forecasting on the basis of fragmentary data, and speech modeling. This is because these mathematical objects either have a simple structure (in the case of wavelets) and/or possess special properties (as orthogonal polynomials do) that can be exploited to decompose complex problems of the types just mentioned into simpler, more manageable ones. For instance, a family of wavelets is obtained by scaling and shifting one particular carefully chosen function, which makes it extremely useful for studying images at various resolutions. Taking advantage of orthogonality, on the other hand, is the most practical way to retrieve information that has a "prescribed direction." In order for wavelets and orthogonal polynomials to be truly useful, however, one must know how to build these functions and must understand their properties as fully as possible. The object of the basic research described above is to contribute to the achievement of both these goals.