A Gaussian quadrature rule for oscillatory integrals on a bounded interval

Andreas Asheim; Alfredo Deaño; Daan Huybrechs; Haiyong Wang

doi:10.3934/dcds.2014.34.883

Article Contents

2014, Volume 34, Issue 3: 883-901. Doi: 10.3934/dcds.2014.34.883

This issue Previous Article Preface Next Article Computing of B-series by automatic differentiation

A Gaussian quadrature rule for oscillatory integrals on a bounded interval

1.
Dept. Computer Science, University of Leuven, Belgium, BE-3001 Heverlee

Received: October 31, 2012

Revised: March 31, 2013

Published: February 2014

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Abstract

We investigate a Gaussian quadrature rule and the corresponding orthogonal polynomials for the oscillatory weight function $e^{i\omega x}$ on the interval $[-1,1]$. We show that such a rule attains high asymptotic order, in the sense that the quadrature error quickly decreases as a function of the frequency $\omega$. However, accuracy is maintained for all values of $\omega$ and in particular the rule elegantly reduces to the classical Gauss-Legendre rule as $\omega \to 0$. The construction of such rules is briefly discussed, and though not all orthogonal polynomials exist, it is demonstrated numerically that rules with an even number of points are well defined. We show that these rules are optimal both in terms of asymptotic order as well as in terms of polynomial order.

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Mathematics Subject Classification: Primary: 65D30, 33C47; Secondary: 65D32, 34E05.

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