## Product integration over infinite intervals. I. Rules based on the zeros of Hermite polynomials

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- by William E. Smith, Ian H. Sloan and Alex H. Opie PDF
- Math. Comp.
**40**(1983), 519-535 Request permission

## Abstract:

The paper discusses both theoretical properties and practical implementation of product integration rules of the form \[ \int _{ - \infty }^\infty {k(x)f(x) dx \approx \sum \limits _{i = 1}^n {{w_{ni}}f({x_{ni}}),} } \] where*f*is continuous,

*k*is absolutely integrable, the nodes $\{ {x_{ni}}\}$ are roots of the Hermite polynomials ${H_n}(x)$, and the weights $\{ {w_{ni}}\}$ are chosen so that the rule is exact if

*f*is any polynomial of degree $< n$. Convergence of the rule to the exact integral as $n \to \infty$ is proved for a wide class of functions

*f*and

*k*(including singular or oscillatory functions

*k*), and rates of convergence are estimated. The rules are shown to have the property of asymptotic positivity, and as a consequence exhibit good numerical stability. Numerical calculations for some practical cases are presented, which show the method to be computationally effective for integrands (including highly oscillatory ones) that decay suitably at infinity. Applications of the method to integration over $[0,\infty )$ are also discussed.

## References

- Milton Abramowitz and Irene A. Stegun,
*Handbook of mathematical functions with formulas, graphs, and mathematical tables*, National Bureau of Standards Applied Mathematics Series, No. 55, U. S. Government Printing Office, Washington, D.C., 1964. For sale by the Superintendent of Documents. MR**0167642**
N. S. Bakhvalov & L. G. Vasil’eva, "Evaluation of the integrals of oscillating functions by interpolation at nodes of Gaussian quadratures," - M. Blakemore, G. A. Evans, and John Hyslop,
*Comparison of some methods for evaluating infinite range oscillatory integrals*, J. Comput. Phys.**22**(1976), no. 3, 352–376. MR**455300**, DOI 10.1016/0021-9991(76)90054-1 - C. W. Clenshaw,
*A note on the summation of Chebyshev series*, Math. Tables Aids Comput.**9**(1955), 118–120. MR**71856**, DOI 10.1090/S0025-5718-1955-0071856-0 - David Elliott and D. F. Paget,
*Product-integration rules and their convergence*, Nordisk Tidskr. Informationsbehandling (BIT)**16**(1976), no. 1, 32–40. MR**405809**, DOI 10.1007/bf01940775 - David Elliott and D. F. Paget,
*The convergence of product integration rules*, BIT**18**(1978), no. 2, 137–141. MR**483319**, DOI 10.1007/BF01931690 - Géza Freud,
*A contribution to the problem of weighted polynomial approximation*, Linear operators and approximation (Proc. Conf., Math. Res. Inst., Oberwolfach, 1971) Internat. Ser. Numer. Math., Vol. 20, Birkhäuser, Basel, 1972, pp. 431–447. MR**0402358**
G. Freud, - Bruno Gabutti,
*On high precision methods for computing integrals involving Bessel functions*, Math. Comp.**33**(1979), no. 147, 1049–1057. MR**528057**, DOI 10.1090/S0025-5718-1979-0528057-5
I. S. Gradshteyn & I. M. Ryzhik, - Gene H. Golub and John H. Welsch,
*Calculation of Gauss quadrature rules*, Math. Comp. 23 (1969), 221-230; addendum, ibid.**23**(1969), no. 106, loose microfiche suppl, A1–A10. MR**0245201**, DOI 10.1090/S0025-5718-69-99647-1 - D. R. Lehman, William C. Parke, and L. C. Maximon,
*Numerical evaluation of integrals containing a spherical Bessel function by product integration*, J. Math. Phys.**22**(1981), no. 7, 1399–1413. MR**626130**, DOI 10.1063/1.525061 - Benjamin Muckenhoupt,
*Mean convergence of Hermite and Laguerre series. I, II*, Trans. Amer. Math. Soc. 147 (1970), 419-431; ibid.**147**(1970), 433–460. MR**0256051**, DOI 10.1090/S0002-9947-1970-0256051-9 - Paul G. Nevai,
*Mean convergence of Lagrange interpolation. II*, J. Approx. Theory**30**(1980), no. 4, 263–276. MR**616953**, DOI 10.1016/0021-9045(80)90030-1
D. F. Paget, - T. N. L. Patterson,
*On high precision methods for the evaluation of Fourier integrals with finite and infinite limits*, Numer. Math.**27**(1976/77), no. 1, 41–52. MR**433932**, DOI 10.1007/BF01399083 - Ian H. Sloan,
*On the numerical evaluation of singular integrals*, BIT**18**(1978), no. 1, 91–102. MR**501799**, DOI 10.1007/BF01947747 - Ian H. Sloan,
*On choosing the points in product integration*, J. Math. Phys.**21**(1980), no. 5, 1032–1039. MR**574876**, DOI 10.1063/1.524552 - Ian H. Sloan and W. E. Smith,
*Product-integration with the Clenshaw-Curtis and related points. Convergence properties*, Numer. Math.**30**(1978), no. 4, 415–428. MR**494863**, DOI 10.1007/BF01398509 - Ian H. Sloan and William E. Smith,
*Product integration with the Clenshaw-Curtis points: implementation and error estimates*, Numer. Math.**34**(1980), no. 4, 387–401. MR**577405**, DOI 10.1007/BF01403676 - Ian H. Sloan and William E. Smith,
*Properties of interpolatory product integration rules*, SIAM J. Numer. Anal.**19**(1982), no. 2, 427–442. MR**650061**, DOI 10.1137/0719027 - William E. Smith and Ian H. Sloan,
*Product-integration rules based on the zeros of Jacobi polynomials*, SIAM J. Numer. Anal.**17**(1980), no. 1, 1–13. MR**559455**, DOI 10.1137/0717001
G. Szegö, - J. V. Uspensky,
*On the convergence of quadrature formulas related to an infinite interval*, Trans. Amer. Math. Soc.**30**(1928), no. 3, 542–559. MR**1501444**, DOI 10.1090/S0002-9947-1928-1501444-8 - Andrew Young,
*Approximate product-integration*, Proc. Roy. Soc. London Ser. A**224**(1954), 552–561. MR**63778**, DOI 10.1098/rspa.1954.0179

*U.S.S.R. Comput. Math. and Math. Phys.*, v. 8, 1968, pp. 241-249.

*Orthogonal Polynomials*, Pergamon Press, Oxford, 1971, Problem 1, pp. 130-131.

*Table of Integrals Series and Products*, Academic Press, New York, 1965.

*Generalised Product Integration*, Ph.D. thesis, University of Tasmania, 1976.

*Orthogonal Polynomials*, Amer. Math. Soc. Colloq. Publ., Vol. 23, Amer. Math. Soc., Providence, R. I., 1939.

## Additional Information

- © Copyright 1983 American Mathematical Society
- Journal: Math. Comp.
**40**(1983), 519-535 - MSC: Primary 65D32
- DOI: https://doi.org/10.1090/S0025-5718-1983-0689468-1
- MathSciNet review: 689468