# American Institute of Mathematical Sciences

March  2014, 34(3): 883-901. doi: 10.3934/dcds.2014.34.883

## A Gaussian quadrature rule for oscillatory integrals on a bounded interval

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

Received  November 2012 Revised  April 2013 Published  August 2013

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.
Citation: Andreas Asheim, Alfredo Deaño, Daan Huybrechs, Haiyong Wang. A Gaussian quadrature rule for oscillatory integrals on a bounded interval. Discrete and Continuous Dynamical Systems, 2014, 34 (3) : 883-901. doi: 10.3934/dcds.2014.34.883
##### References:
 [1] A. Asheim and D. Huybrechs, Gaussian quadrature for oscillatory integral transforms, IMA J. Numer. Anal, (2013). doi: 10.1093/imanum/drs060. [2] P. Bleher and A. Its, Asymptotics of the partition function of a random matrix model, in "Ann. Inst. Fourier," 55 (2005), 1943-2000. doi: 10.5802/aif.2147. [3] L. Filon, On a quadrature formula for trigonometric integrals, Proc. Roy. Soc. Edinburgh, 49 (1928), 38-47. [4] W. Gautschi, "Orthogonal Polynomials: Computation and Approximation," Oxford University Press, 2004. [5] D. Huybrechs and S. Olver, Superinterpolation in highly oscillatory quadrature, Found. Comput. Math, 12 (2012), 203-228. doi: 10.1007/s10208-011-9102-8. [6] D. Huybrechs and S. Vandewalle, On the evaluation of highly oscillatory integrals by analytic continuation, SIAM J. Numer. Anal., 44 (2006), 1026-1048. doi: 10.1137/050636814. [7] A. Iserles, Think globally, act locally: Solving highly-oscillatory ordinary differential equations, Appl. Numer. Math., 43 (2002), 145-160. doi: 10.1016/S0168-9274(02)00122-8. [8] A. Iserles, On the numerical quadrature of highly-oscillating integrals I: Fourier transforms, IMA J. Numer. Anal., 24 (2004), 365-391. doi: 10.1093/imanum/24.3.365. [9] A. Iserles and S. Nørsett, Efficient quadrature of highly oscillatory integrals using derivatives, Proc. R. Soc. A, 461 (2005), 1383-1399. doi: 10.1098/rspa.2004.1401. [10] A. Iserles and S. P. Nørsett, On quadrature methods for highly oscillatory integrals and their implementation, BIT, 44 (2004), 755-772. doi: 10.1007/s10543-004-5243-3. [11] A. Iserles and S. P. Nørsett, On the computation of highly oscillatory multivariate integrals with stationary points, BIT, 46 (2006), 549-566. doi: 10.1007/s10543-006-0071-2. [12] A. Iserles and S. P. Nørsett, Quadrature methods for multivariate highly oscillatory integrals using derivatives, Math. Comp., 75 (2006), 1233-1258. doi: 10.1090/S0025-5718-06-01854-0. [13] L. G. Ixaru and B. Paternoster, A Gauss quadrature rule for oscillatory integrands, Comput. Phys. Commun., 133 (2001), 177-188. doi: 10.1016/S0010-4655(00)00173-9. [14] V. Ledoux and M. Van Daele, Interpolatory quadrature rules for oscillatory integrals, J. Sci. Comput., 53 (2012), 586-607. doi: 10.1007/s10915-012-9589-4. [15] D. Levin, Fast integration of rapidly oscillatory functions, J. Comput. Appl. Math., 67 (1996), 95-101. doi: 10.1016/0377-0427(94)00118-9. [16] J. L. López and N. M. Temme, Two-point Taylor expansions of analytic functions, Stud. Appl. Math., 109 (2002), 297-311. doi: 10.1111/1467-9590.00225. [17] F. Olver, D. Lozier, R. Boisvert and C. Clark, "NIST Handbook of Mathematical Functions," Cambridge University Press, 2010. [18] S. Olver, Moment-free numerical integration of highly oscillatory functions, IMA J. Numer. Anal., 26 (2006), 213-227. doi: 10.1093/imanum/dri040. [19] S. Olver, Fast, numerically stable computation of oscillatory integrals with stationary points, BIT, 50 (2010), 149-171. doi: 10.1007/s10543-010-0251-y.

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##### References:
 [1] A. Asheim and D. Huybrechs, Gaussian quadrature for oscillatory integral transforms, IMA J. Numer. Anal, (2013). doi: 10.1093/imanum/drs060. [2] P. Bleher and A. Its, Asymptotics of the partition function of a random matrix model, in "Ann. Inst. Fourier," 55 (2005), 1943-2000. doi: 10.5802/aif.2147. [3] L. Filon, On a quadrature formula for trigonometric integrals, Proc. Roy. Soc. Edinburgh, 49 (1928), 38-47. [4] W. Gautschi, "Orthogonal Polynomials: Computation and Approximation," Oxford University Press, 2004. [5] D. Huybrechs and S. Olver, Superinterpolation in highly oscillatory quadrature, Found. Comput. Math, 12 (2012), 203-228. doi: 10.1007/s10208-011-9102-8. [6] D. Huybrechs and S. Vandewalle, On the evaluation of highly oscillatory integrals by analytic continuation, SIAM J. Numer. Anal., 44 (2006), 1026-1048. doi: 10.1137/050636814. [7] A. Iserles, Think globally, act locally: Solving highly-oscillatory ordinary differential equations, Appl. Numer. Math., 43 (2002), 145-160. doi: 10.1016/S0168-9274(02)00122-8. [8] A. Iserles, On the numerical quadrature of highly-oscillating integrals I: Fourier transforms, IMA J. Numer. Anal., 24 (2004), 365-391. doi: 10.1093/imanum/24.3.365. [9] A. Iserles and S. Nørsett, Efficient quadrature of highly oscillatory integrals using derivatives, Proc. R. Soc. A, 461 (2005), 1383-1399. doi: 10.1098/rspa.2004.1401. [10] A. Iserles and S. P. Nørsett, On quadrature methods for highly oscillatory integrals and their implementation, BIT, 44 (2004), 755-772. doi: 10.1007/s10543-004-5243-3. [11] A. Iserles and S. P. Nørsett, On the computation of highly oscillatory multivariate integrals with stationary points, BIT, 46 (2006), 549-566. doi: 10.1007/s10543-006-0071-2. [12] A. Iserles and S. P. Nørsett, Quadrature methods for multivariate highly oscillatory integrals using derivatives, Math. Comp., 75 (2006), 1233-1258. doi: 10.1090/S0025-5718-06-01854-0. [13] L. G. Ixaru and B. Paternoster, A Gauss quadrature rule for oscillatory integrands, Comput. Phys. Commun., 133 (2001), 177-188. doi: 10.1016/S0010-4655(00)00173-9. [14] V. Ledoux and M. Van Daele, Interpolatory quadrature rules for oscillatory integrals, J. Sci. Comput., 53 (2012), 586-607. doi: 10.1007/s10915-012-9589-4. [15] D. Levin, Fast integration of rapidly oscillatory functions, J. Comput. Appl. Math., 67 (1996), 95-101. doi: 10.1016/0377-0427(94)00118-9. [16] J. L. López and N. M. Temme, Two-point Taylor expansions of analytic functions, Stud. Appl. Math., 109 (2002), 297-311. doi: 10.1111/1467-9590.00225. [17] F. Olver, D. Lozier, R. Boisvert and C. Clark, "NIST Handbook of Mathematical Functions," Cambridge University Press, 2010. [18] S. Olver, Moment-free numerical integration of highly oscillatory functions, IMA J. Numer. Anal., 26 (2006), 213-227. doi: 10.1093/imanum/dri040. [19] S. Olver, Fast, numerically stable computation of oscillatory integrals with stationary points, BIT, 50 (2010), 149-171. doi: 10.1007/s10543-010-0251-y.
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