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 | References | Similar Articles | Additional Information
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.
- A. Erdélyi, Asymptotic representations of Fourier integrals and the method of stationary phase, J. Soc. Indust. Appl. Math. 3 (1955), 17–27. MR 70744
- M. J. Lighthill, Introduction to Fourier analysis and generalised functions, Cambridge Monographs on Mechanics and Applied Mathematics, Cambridge University Press, New York, 1958. MR 0092119
- J. N. Lyness and B. W. Ninham, Numerical quadrature and asymptotic expansions, Math. Comp. 21 (1967), 162–178. MR 225488, DOI https://doi.org/10.1090/S0025-5718-1967-0225488-X
- 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, DOI https://doi.org/10.1090/S0025-5718-1970-0260230-8
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Additional Information
Keywords:
Fourier coefficients,
Euler-Maclaurin summation formula,
Fourier coefficient asymptotic expansion,
numerical quadrature,
subtracting out singularities
Article copyright:
© Copyright 1971
American Mathematical Society