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Adjusted forms of the Fourier coefficient asymptotic expansion and applications in numerical quadrature


Author: J. N. Lyness
Journal: Math. Comp. 25 (1971), 87-104
MSC: Primary 42.10
DOI: https://doi.org/10.1090/S0025-5718-1971-0290020-2
MathSciNet review: 0290020
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Abstract: The conventional Fourier coefficient asymptotic expansion is derived by means of a specific contour integration. An adjusted expansion is obtained by deforming this contour. A corresponding adjustment to the Euler-Maclaurin expansion exists. The effect of this adjustment in the error functional for a general quadrature rule is investigated. It is the same as the effect of subtracting out a pair of complex poles from the integrand, using an unconventional subtraction function. In certain applications, the use of this subtraction function is of practical value.

An incidental result is a direct proof of Erdélyi's formula for the Fourier coefficient asymptotic expansion, valid when $ f(x)$ has algebraic or logarithmic singularities, but is otherwise analytic.


References [Enhancements On Off] (What's this?)

  • [1] A. Erdélyi, "Asymptotic representations of Fourier integrals and the method of stationary phase," J. Soc. Indust. Appl. Math., v. 3, 1955, pp. 17-27. MR 17, 29. MR 0070744 (17:29g)
  • [2] M. J. Lighthill, Introduction to Fourier Analysis and Generalized Functions, Cambridge Univ. Press, New York, 1958; 1960. MR 19, 1066; MR 22 #5888. MR 0092119 (19:1066a)
  • [3] 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)
  • [4] 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)

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1971-0290020-2
Keywords: Fourier coefficients, Euler-Maclaurin summation formula, Fourier coefficient asymptotic expansion, numerical quadrature, subtracting out singularities
Article copyright: © Copyright 1971 American Mathematical Society

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