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 has algebraic or logarithmic singularities, but is otherwise analytic.

**[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