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

Author:
J. N. Lyness

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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

**[1]***Handbook of mathematical functions, with formulas, graphs, and mathematical tables*, Edited by Milton Abramowitz and Irene A. Stegun, Dover Publications, Inc., New York, 1966. MR**0208797****[2]**W. G. Bickley,*Formulae for numerical differentiation*, Math. Gaz.**25**(1941), 19–27. MR**0003580**, https://doi.org/10.2307/3606475**[3]**James W. Cooley and John W. Tukey,*An algorithm for the machine calculation of complex Fourier series*, Math. Comp.**19**(1965), 297–301. MR**0178586**, https://doi.org/10.1090/S0025-5718-1965-0178586-1**[4]**Philip J. Davis and Philip Rabinowitz,*Numerical integration*, Blaisdell Publishing Co. Ginn and Co., Waltham, Mass.-Toronto, Ont.-London, 1967. MR**0211604****[5]**L. N. G. Filon, "On a quadrature formula for trigonometric integrals,"*Proc. Roy. Soc. Edinburgh*, v. 49, 1929, pp. 38-47.**[6]**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.**[7]**Richard R. Goldberg and Richard S. Varga,*Moebius inversion of Fourier transforms*, Duke Math. J.**23**(1956), 553–559. MR**0080800****[8]**R. W. Hamming,*Numerical methods for scientists and engineers*, 2nd ed., McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. International Series in Pure and Applied Mathematics. MR**0351023****[9]**G. H. Hardy and E. M. Wright,*An introduction to the theory of numbers*, Oxford, at the Clarendon Press, 1954. 3rd ed. MR**0067125****[10]**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****[11]**E. Landau,*Vorlesungen über Zahlentheorie.*Band II, Chelsea, New York, 1947.**[12]**Yudell L. Luke,*On the computation of oscillatory integrals*, Proc. Cambridge Philos. Soc.**50**(1954), 269–277. MR**0062518****[13]**J. N. Lyness,*Quadrature methods based on complex function values*, Math. Comp.**23**(1969), 601–619. MR**0247771**, https://doi.org/10.1090/S0025-5718-1969-0247771-6**[14]**J. N. Lyness and C. B. Moler,*Generalized Romberg methods for integrals of derivatives*, Numer. Math.**14**(1969/1970), 1–13. MR**0256564**, https://doi.org/10.1007/BF02165095**[15]**L. M. Milne-Thompson,*The Calculus of Finite Differences*, Macmillan, London, 1933.**[16]**C. Ballester and V. Pereyra,*On the construction of discrete approximations to linear differential expressions*, Math. Comp.**21**(1967), 297–302. MR**0228167**, https://doi.org/10.1090/S0025-5718-1967-0228167-8

Retrieve articles in *Mathematics of Computation*
with MSC:
65.90

Retrieve articles in all journals with MSC: 65.90

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1970-0260230-8

Keywords:
Fourier coefficients,
Fourier series,
Poisson summation formula,
Euler-Maclaurin summation formula,
Möbius inversion,
Filon-Luke formulas,
fast Fourier transform,
trapezoidal quadrature rule,
Fourier coefficient asymptotic expansion

Article copyright:
© Copyright 1970
American Mathematical Society