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
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