The calculation of Fourier coefficients by the Möbius inversion of the Poisson summation formula. I. Functions whose early derivatives are continuous

Lyness, J. N.

doi:10.1090/S0025-5718-1970-0260230-8

The calculation of Fourier coefficients by the Möbius inversion of the Poisson summation formula. I. Functions whose early derivatives are continuous
HTML articles powered by AMS MathViewer

by J. N. Lyness PDF

Math. Comp. 24 (1970), 101-135 Request permission

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.

References

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

Proc. Roy. Soc. Edinburgh

Fast Fourier Transforms for Fun and Profit

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

Vorlesungen über Zahlentheorie.

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

The Calculus of Finite Differences

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