Mathematical Sciences: Pade and Other Rational Approximation; Moment Theory

这个数学研究项目将集中于一个复变量的复值函数的近似。工作的中心是利用帕德近似的合理插值。这些是有理函数，它们与特定函数在两个或多个点上的幂级数相匹配。在这个框架内，将研究不确定强Stieltjes矩问题和相关的正交Laurent多项式，Szego多项式在数字信号处理中的应用，以及相对于经典特殊函数的Pade连分式的收敛性，包括截断误差分析。最后，对数学函数的连分式表示的全面调查以及它们的计算指南将准备。大规模计算将用于寻找可证明的一般性质的实验，以及有理近似收敛行为的图形和数值说明。在许多科学应用中，相关函数是不能用初等方法计算的。近似理论，正如这个项目所例证的那样，寻求建立和分类方法，通过特定的，易于计算的替代品来近似或多或少的一般函数。重要的是，所开发的方法还提供误差分析，并在可能的情况下提供有效实现近似的算法。＊＊＊

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. ***