A Gaussian quadrature rule for oscillatory integrals on a bounded interval

Abstract

We investigate a Gaussian quadrature rule and the corresponding orthogonal polynomials for the oscillatory weight function $e^{iωx}$ 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 $ω$. However, accuracy is maintained for all values of $ω$ and in particular the rule elegantly reduces to the classical Gauss-Legendre rule as $ω\to 0$. 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 always well defined. We show that these rules are optimal both in terms of asymptotic order as well as in terms of polynomial order.