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

Lyness, J. N.

doi:10.1090/S0025-5718-1971-0290020-2

Adjusted forms of the Fourier coefficient asymptotic expansion and applications in numerical quadrature
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by J. N. Lyness PDF

Math. Comp. 25 (1971), 87-104 Request permission

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

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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 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 10.1090/S0025-5718-1970-0260230-8