Gaussian quadrature is a powerful technique for numerical integration that falls under the broad category of spectral methods. The purpose of this work is to provide an introduction to the theory and practice of Gaussian quadrature. We study the approximation theory of trigonometric and orthogonal polynomials and related functions and examine the analytical framework of Gaussian quadrature. We discuss Gaussian quadrature for bandlimited functions, a topic inspired by some recent developments in the analysis of prolate spheroidal wave functions. Algorithms for the computation of the quadrature nodes and weights are described. Several applications of Gaussian quadrature are given, ranging from the evaluation of special functions to pseudospectral methods for solving differential equations. Software realization of select algorithms is provided. Table of Contents: Introduction / Approximating with Polynomials and Related Functions / Gaussian Quadrature / Applications / Links to Mathematical Software
Introduction.- Approximating with Polynomials and Related Functions.- Gaussian Quadrature.- Applications.- Links to Mathematical Software.
Narayan Kovvali received the B.Tech. degree in electrical engineering from the Indian Institute of Technology, Kharagpur, India, in 2000, and the M.S. and Ph.D. degrees in electrical engineering from Duke University, Durham, North Carolina, in 2002 and 2005, respectively. In 2006, he joined the Department of Electrical Engineering at Arizona State University, Tempe, Arizona, as Assistant Research Scientist. He currently holds the position of Assistant Research Professor in the School of Electrical, Computer, and Energy Engineering at Arizona State University. His research interests include statistical signal processing, detection, estimation, stochastic filtering and tracking, Bayesian data analysis, multi-sensor data fusion, Monte Carlo methods, and scientific computing. Dr. Kovvali is a Senior Member of the IEEE.
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