The calculation of Fourier coefficients by the Möbius inversion of the Poisson summation formula. II. Piecewise continuous functions and functions with poles near the interval
Abstract: In Part I, the MIPS method for calculating Fourier coefficients was introduced, and applied to functions . In this part two extensions of the theory are described.
One modification extends the theory to piecewise continuous functions, . Using these results the method may be used to calculate approximations to trigonometrical integrals (in which the length of the interval need not coincide with a period of the trigonometrical weighting function).
The other modification treats functions which are analytic, but whose low-order derivatives vary rapidly due to poles in the complex plane near the interval of integration. Essentially these poles are 'subtracted out' but this is done implicitly by the inclusion of additional terms in the standard series.
The practical application of these modified methods requires that the nature and location of the discontinuities--or poles--be known at least approximately.
-  Milton Abramowitz and Irene A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, vol. 55, For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1964. MR 0167642
-  J. N. Lyness, The calculation of Fourier coefficients by the Möbius inversion of the Poisson summation formula. I. Functions whose early derivatives are continuous, Math. Comp. 24 (1970), 101–135. MR 260230, https://doi.org/10.1090/S0025-5718-1970-0260230-8
-  J. N. Lyness, Adjusted forms of the Fourier coefficient asymptotic expansion and applications in numerical quadrature, Math. Comp. 25 (1971), 87–104. MR 290020, https://doi.org/10.1090/S0025-5718-1971-0290020-2
-  J. N. Lyness and B. W. Ninham, Numerical quadrature and asymptotic expansions, Math. Comp. 21 (1967), 162–178. MR 225488, https://doi.org/10.1090/S0025-5718-1967-0225488-X
-  W. E. Smith and J. N. Lyness, Applications of Hilbert transform theory to numerical quadrature, Math. Comp. 23 (1969), 231–252. MR 251906, https://doi.org/10.1090/S0025-5718-1969-0251906-9
- M. Abramowitz & I. A. Stegun, (Editors), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Nat. Bur. Standards Appl. Math. Series, 55, Superintendent of Documents, U. S. Government Printing Office, Washington, D. C., 1964; 3rd printing with corrections, 1965. MR 29 #4914; MR 31 #1400. MR 0167642 (29:4914)
- J. N. Lyness, "The calculation of Fourier coefficients by the Möbius inversion of the Poisson summation formula. I," Math. Comp., v. 24, 1970, pp. 101-135. MR 0260230 (41:4858)
- J. N. Lyness, "Adjusted forms of the Fourier coefficient asymptotic expansion and applications in numerical quadrature," Math. Comp., v. 25, 1971, pp. 87-104. MR 0290020 (44:7205)
- J. N. Lyness, & B. W. Ninham, "Numerical quadrature and asymptotic expansions," Math. Comp., v. 21, 1967, pp. 162-178. MR 37 #1081. MR 0225488 (37:1081)
- W. E. Smith & J. N. Lyness, "Applications of Hilbert transform theory to numerical quadrature," Math. Comp., v. 23, 1969, pp. 231-252. MR 40 #5132. MR 0251906 (40:5132)
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Keywords: Fourier coefficients, Euler-Maclaurin summation formula, Fourier coefficient asymptotic expansion, numerical quadrature, subtracting out singularities
Article copyright: © Copyright 1971 American Mathematical Society