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 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.**3**(1955), 17–27. MR**0070744****[2]**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****[3]**J. N. Lyness and B. W. Ninham,*Numerical quadrature and asymptotic expansions*, Math. Comp.**21**(1967), 162–178. MR**0225488**, https://doi.org/10.1090/S0025-5718-1967-0225488-X**[4]**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**0260230**, https://doi.org/10.1090/S0025-5718-1970-0260230-8

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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