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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
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.

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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