Sinc function quadrature rules for the Fourier integral
Author:
John Lund
Journal:
Math. Comp. 41 (1983), 103113
MSC:
Primary 65D30
MathSciNet review:
701627
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Abstract: In this paper a numerical algorithm is proposed for the computation of the Fourier Transform. The quadrature rule developed is based on the Whittaker Cardinal Function expansion of the integrand and a certain Conformal Map. The error of the method is analyzed and numerical results are reported which confirm the accuracy of the quadrature rule.
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 [1]
 T. S. Bromwich, An Introduction to the Theory of Infinite Series, Macmillan, New York, 1966.
 [2]
 G. Debalbine & J. N. Franklin, "The calculation of Fourier Integrals," Math. Comp., v. 20, 1966, pp. 570589. MR 0203976 (34:3823)
 [3]
 L. N. Filon, "On a quadrature formula for trigonometric integrals," Proc. Roy. Soc. Edinburgh, v. 49, 1929, pp. 3847.
 [4]
 M. Murwitz, Jr. & P. F. Zweifel, "Numerical quadrature of Fourier transform integrals," MTAC, v. 10, 1956, pp. 140149. MR 0080994 (18:337h)
 [5]
 V. I. Krylov & N. S. Skoblya, Handbook of Numerical Inversion of Laplace Transforms, Israel Program for Sci. Transl., Jerusalem, 1969. MR 0391481 (52:12302)
 [6]
 Y. L. Luke, "On the computation of oscillatory integrals," Proc. Cambridge Philos. Soc., v. 50, 1954, pp. 269277. MR 0062518 (15:992b)
 [7]
 M. K. Miller & W. T. Guy, Jr., "Numerical inversion of the Laplace transform by use of Jacobi polynomials," SIAM J. Numer. Anal., v. 3, 1966, pp. 624635. MR 0212995 (35:3860)
 [8]
 F. W. J. Olver, Introduction to Asymptotics and Special Functions, Academic Press, New York, 1974. MR 0435697 (55:8655)
 [9]
 M. E. Salzer, "Orthogonal polynomials arising in the numerical evaluations of inverse Laplace transforms," MTAC, v. 9, 1955, pp. 164177. MR 0078498 (17:1203d)
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 F. Stenger, "Integration rules via the trapezoid formula," J. Inst. Math. Appl., v. 12, 1973, pp. 103114.
 [11]
 F. Stenger, "Numerical methods based on Whittaker cardinal, or sinc functions," SIAM Rev., v. 23, 1981, pp. 165224. MR 618638 (83g:65027)
 [12]
 A. J. van de Vooren & M. J. van Linde, "Numerical calculation of integrals with strongly oscillating integrands," Math. Comp., v. 20, 1966, pp. 232245. MR 0192644 (33:869)
 [13]
 G. Walter & D. Schultz, "Some eigenfunction methods for computing a numerical Fourier transform," J. Inst. Math. Appl., v. 18, 1976, pp. 279293. MR 0455505 (56:13743)
 [14]
 M. Weber, "Numerical computation of the Fourier transform using Laguerre functions and the fast Fourier transform," Numer. Math., v. 36, 1981, pp. 197209. MR 611492 (82c:65095)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198307016278
PII:
S 00255718(1983)07016278
Article copyright:
© Copyright 1983
American Mathematical Society
