Abstract:The Möbius inversion technique is applied to the Poisson summation formula. This results in expressions for the remainder term in the Fourier coefficient asymptotic expansion as an infinite series. Each element of this series is a remainder term in the corresponding Euler-Maclaurin summation formula, and the series has specified convergence properties. These expressions may be used as the basis for the numerical evaluation of sets of Fourier coefficients. The organization of such a calculation is described, and discussed in the context of a broad comparison between this approach and various other standard methods.
- Milton Abramowitz and Irene A. Stegun (eds.), Handbook of mathematical functions, with formulas, graphs, and mathematical tables, Dover Publications, Inc., New York, 1966. MR 0208797
- W. G. Bickley, Formulae for numerical differentiation, Math. Gaz. 25 (1941), 19–27. MR 3580, DOI 10.2307/3606475
- James W. Cooley and John W. Tukey, An algorithm for the machine calculation of complex Fourier series, Math. Comp. 19 (1965), 297–301. MR 178586, DOI 10.1090/S0025-5718-1965-0178586-1
- Philip J. Davis and Philip Rabinowitz, Numerical integration, Blaisdell Publishing Co. [Ginn and Co.], Waltham, Mass.-Toronto, Ont.-London, 1967. MR 0211604 L. N. G. Filon, "On a quadrature formula for trigonometric integrals," Proc. Roy. Soc. Edinburgh, v. 49, 1929, pp. 38–47. W. M. Gentleman & G. Sande, Fast Fourier Transforms for Fun and Profit, Proc. AFIPS 1966 Fall Joint Computer Conf., v. 29, 1966, pp. 563–578.
- Richard R. Goldberg and Richard S. Varga, Moebius inversion of Fourier transforms, Duke Math. J. 23 (1956), 553–559. MR 80800
- R. W. Hamming, Numerical methods for scientists and engineers, 2nd ed., International Series in Pure and Applied Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. MR 0351023
- G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, Oxford, at the Clarendon Press, 1954. 3rd ed. MR 0067125
- Zdeněk Kopal, Numerical analysis. With emphasis on the application of numerical techniques to problems of infinitesimal calculus in single variable, John Wiley & Sons, Inc., New York, 1955. MR 0077213 E. Landau, Vorlesungen über Zahlentheorie. Band II, Chelsea, New York, 1947.
- Yudell L. Luke, On the computation of oscillatory integrals, Proc. Cambridge Philos. Soc. 50 (1954), 269–277. MR 62518, DOI 10.1017/S0305004100029327
- J. N. Lyness, Quadrature methods based on complex function values, Math. Comp. 23 (1969), 601–619. MR 247771, DOI 10.1090/S0025-5718-1969-0247771-6
- J. N. Lyness and C. B. Moler, Generalized Romberg methods for integrals of derivatives, Numer. Math. 14 (1969/70), 1–13. MR 256564, DOI 10.1007/BF02165095 L. M. Milne-Thompson, The Calculus of Finite Differences, Macmillan, London, 1933.
- C. Ballester and V. Pereyra, On the construction of discrete approximations to linear differential expressions, Math. Comp. 21 (1967), 297–302. MR 228167, DOI 10.1090/S0025-5718-1967-0228167-8
- © Copyright 1970 American Mathematical Society
- Journal: Math. Comp. 24 (1970), 101-135
- MSC: Primary 65.90
- DOI: https://doi.org/10.1090/S0025-5718-1970-0260230-8
- MathSciNet review: 0260230