Sinc function quadrature rules for the Fourier integral

Author:
John Lund

Journal:
Math. Comp. **41** (1983), 103-113

MSC:
Primary 65D30

MathSciNet review:
701627

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Abstract | References | Similar Articles | Additional Information

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.

**[1]**T. S. Bromwich,*An Introduction to the Theory of Infinite Series*, Macmillan, New York, 1966.**[2]**Guy de Balbine and Joel N. Franklin,*The calculation of Fourier integrals*, Math. Comp.**20**(1966), 570–589. MR**0203976**, 10.1090/S0025-5718-1966-0203976-9**[3]**L. N. Filon, "On a quadrature formula for trigonometric integrals,"*Proc. Roy. Soc. Edinburgh*, v. 49, 1929, pp. 38-47.**[4]**H. Hurwitz Jr. and P. F. Zweifel,*Numerical quadrature of Fourier transform integrals*, Math. Tables Aids Comput.**10**(1956), 140–149. MR**0080994**, 10.1090/S0025-5718-1956-0080994-9**[5]**V. I. Krylov and N. S. Skoblya,*Handbook of numerical inversion of Laplace transforms*, Israel Program for Scientific Translations, Jerusalem, 1969. Translated from the Russian by D. Louvish. MR**0391481****[6]**Yudell L. Luke,*On the computation of oscillatory integrals*, Proc. Cambridge Philos. Soc.**50**(1954), 269–277. MR**0062518****[7]**Max K. Miller and W. T. Guy Jr.,*Numerical inversion of the Laplace transform by use of Jacobi polynomials*, SIAM J. Numer. Anal.**3**(1966), 624–635. MR**0212995****[8]**F. W. J. Olver,*Asymptotics and special functions*, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1974. Computer Science and Applied Mathematics. MR**0435697****[9]**Herbert E. Salzer,*Orthogonal polynomials arising in the numerical evaluation of inverse Laplace transforms*, Math. Tables Aids Comput.**9**(1955), 164–177. MR**0078498**, 10.1090/S0025-5718-1955-0078498-1**[10]**F. Stenger, "Integration rules via the trapezoid formula,"*J. Inst. Math. Appl.*, v. 12, 1973, pp. 103-114.**[11]**Frank Stenger,*Numerical methods based on Whittaker cardinal, or sinc functions*, SIAM Rev.**23**(1981), no. 2, 165–224. MR**618638**, 10.1137/1023037**[12]**A. I. van de Vooren and H. J. van Linde,*Numerical calculation of integrals with strongly oscillating integrand*, Math. Comp.**20**(1966), 232–245. MR**0192644**, 10.1090/S0025-5718-1966-0192644-8**[13]**G. Walter and D. Schultz,*Some eigenfunction methods for computing a numerical Fourier transform*, J. Inst. Math. Appl.**18**(1976), no. 3, 279–293. MR**0455505****[14]**H. Weber,*Numerical computation of the Fourier transform using Laguerre functions and the fast Fourier transform*, Numer. Math.**36**(1980/81), no. 2, 197–209. MR**611492**, 10.1007/BF01396758

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DOI:
https://doi.org/10.1090/S0025-5718-1983-0701627-8

Article copyright:
© Copyright 1983
American Mathematical Society