Mathematical Sciences: Pade and Other Rational Approximation; Moment Theory

本项目将重点研究复杂域的有理插值和逼近问题。重点将放在帕德近似和相关的连分式上。为研究收敛性、收敛速度、收敛加速度、截断误差分析、力矩理论和稳定性分析，将发展复分析的建设性方法。数值分析和大规模计算的方法将用于数值实验（以获得对复杂问题的新见解）和特殊函数计算的数值结果。这项工作涉及基本的数学分析，并应用于数字滤波和信号处理、特殊函数理论和非线性Riccati微分方程的解。数字信号处理在天文学、经济学、电力规划、医学、雷达、地震学和语音处理等各个领域都有应用。预计该项目将对理性逼近的现状有重要的补充，并为今后的研究指明新的方向。

This project will focus on problems of rational interpolation and approximation in the complex domain. Emphasis will be placed on Pade approximants and related continued fractions. Constructive methods of complex analysis will be developed for the study of convergence, speed of convergence, acceleration of convergence, truncation error analysis, moment theory and stability analysis. Methods of numerical analysis and large-scale computations will be used both for numerical experimentation (to gain new insight into complex problems) and for numerical results on the computation of special functions. The work involves fundamental mathematical analysis with applications to digital filtering and signal processing, to the theory of special functions and to the solution of nonlinear Riccati differential equations. Digital signal processing has applications to diverse areas such as astronomy, economics, electrical power planning, medicine, radar, seismology and speech processing. It is expected that the project will add significantly to the present state of knowledge of rational approximation and point out new directions for future research.