Mathematical Sciences: Pade and Other Rational Approximation; Moment Theory

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Jones 9302584 This mathematical research project will focus on the approximation of complex valued functions of a complex variable. The work centers on the use of rational interpolation by Pade approximates. These are rational functions which match the power series of specific functions at two or more points. Within this framework investigations will be carried out on the indeterminate strong Stieltjes moment problem and the associated orthogonal Laurent polynomials, on applications of Szego polynomials to digital signal processing and the convergence of Pade continued fractions relative to classical special functions including truncation error analysis. Finally, a comprehensive survey of continued fraction representations of mathematical functions along with a guide to their computation will be prepared. Large scale computation will be done for experiments in search of general properties that can be proven and for graphical and numerical illustration of the convergence behavior of rational approximants. In many scientific applications, the relevant functions are not computable by elementary methods. Approximation theory, as exemplified by this project, seeks to establish and catalogue methods for approximating more or less general functions by means of specific, easily computable substitutes. It is important that the methods developed also provide error analysis and, where possible, algorithms for efficient implementation of the approximations. ***