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A Gaussian quadrature rule for oscillatory integrals on a bounded interval. (English) Zbl 1278.65023

Summary: 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.

MSC:

65D32 Numerical quadrature and cubature formulas
41A55 Approximate quadratures
65T40 Numerical methods for trigonometric approximation and interpolation
42A16 Fourier coefficients, Fourier series of functions with special properties, special Fourier series

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References:

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