Mathematical Sciences: Approximation Theory and Potentials

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DMS-9415657 Totik Totik will investigate several problems in the general areas of approximation theory with special interest in applications of potentials and in polynomials. The main topics include logarithmic potentials in the presence of an external field, polynomial approximation with varying weights, fast decreasing polynomials, asymptotic for orthogonal polynomials, zeros of polynomials, and approximation by positive operators. Harmonic analysis combines those elements of mathematics best exemplifying the ideas of synthesis. One seeks to decompose complex problems into fundamental components. These components are then analyzed for their basic characteristics. Finally, the solution is reconstructed through a recombination of the components. The Fourier series and Fourier transform are examples of tools used in this context; one discrete , the other representing a continuous decomposition. Orthogonal polynomials play an important role in certain segments of harmonic analysis because (a) they arise naturally in the study of differential equations (b) they can be used to represent large classes of functions and (c) they are often easily computable.