A Gaussian quadrature rule for oscillatory integrals on a bounded interval
نویسندگان
چکیده
We investigate a Gaussian quadrature rule and the corresponding orthogonal polynomials for the oscillatory weight function ei!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 !. However, accuracy is maintained for all values of ! and in particular the rule elegantly reduces to the classical Gauss-Legendre rule as ! ! 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.
منابع مشابه
Numerical methods for highly oscillatory integrals on semi-finite intervals
In highly oscillatory integrals, the integrand fw(x) oscillates rapidly with a frequency ω. For very high values of ω, numerical evaluation of such integrals by Gaussian quadrature rules can be of very low accuracy. In such problems which have many applications in mathematical physics, it is important to devise algorithms with errors which decrease as fast as w−N , for some N > 0. In this paper...
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