Quadrature formulae and functions of exponential type

Rahman, Qazi I.; Schmeisser, Gerhard

doi:10.1090/S0025-5718-1990-0990602-8

Quadrature formulae and functions of exponential type
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Math. Comp. 54 (1990), 245-270 Request permission

Abstract:

In this paper we obtain certain generalizations of the trapezoidal rule and the Euler-Maclaurin formula that involve derivatives. In the case of quadrature of functions of exponential type over infinite intervals we find conditions under which existence of the (improper) integral and convergence of the approximating series become equivalent. In the process, we also establish a best possible version of a theorem of R. P. Boas and A. C. Schaeffer.

References

Ralph Philip Boas Jr., Entire functions, Academic Press, Inc., New York, 1954. MR 0068627
R. P. Boas Jr. and A. C. Schaeffer, New inequalities for entire functions, J. Math. Mech. 7 (1958), 191–205. MR 0092856, DOI 10.1512/iumj.1958.7.57016

Identitätssätze für ganze Funktionen vom Exponentialtyp

J. D. Donaldson and David Elliott, A unified approach to quadrature rules with asymptotic estimates of their remainders, SIAM J. Numer. Anal. 9 (1972), 573–602. MR 317522, DOI 10.1137/0709051
Robert Gervais, Qazi Ibadur Rahman, and Gerhard Schmeisser, A bandlimited function simulating a duration-limited one, Anniversary volume on approximation theory and functional analysis (Oberwolfach, 1983) Internat. Schriftenreihe Numer. Math., vol. 65, Birkhäuser, Basel, 1984, pp. 355–362. MR 820536
G. H. Hardy, Divergent Series, Oxford, at the Clarendon Press, 1949. MR 0030620
Peter Henrici, Applied and computational complex analysis, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. Volume 1: Power series—integration—conformal mapping—location of zeros. MR 0372162
R. Kreß, On the general Hermite cardinal interpolation, Math. Comp. 26 (1972), 925–933. MR 320586, DOI 10.1090/S0025-5718-1972-0320586-6

Le calcul des résidus

John McNamee, Error-bounds for the evaluation of integrals by the Euler-Maclaurin formula and by Gauss-type formulae, Math. Comp. 18 (1964), 368–381. MR 185804, DOI 10.1090/S0025-5718-1964-0185804-1
P. Olivier and Q. I. Rahman, Sur une formule de quadrature pour des fonctions entières, RAIRO Modél. Math. Anal. Numér. 20 (1986), no. 3, 517–537 (French, with English summary). MR 862790, DOI 10.1051/m2an/1986200305171

Praktische Mathematik

Frank Stenger, Integration formulae based on the trapezoidal formula, J. Inst. Math. Appl. 12 (1973), 103–114. MR 381261
Frank Stenger, Numerical methods based on Whittaker cardinal, or sinc functions, SIAM Rev. 23 (1981), no. 2, 165–224. MR 618638, DOI 10.1137/1023037
Hidetosi Takahasi and Masatake Mori, Estimation of errors in the numerical quadrature of analytic functions, Applicable Anal. 1 (1971), no. 3, 201–229. MR 288960, DOI 10.1080/00036817108839015

The theory of functions