Adjusted forms of the Fourier coefficient asymptotic expansion and applications in numerical quadrature
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
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.
M. J. Lighthill, Introduction to Fourier Analysis and Generalized Functions, Cambridge Univ. Press, New York, 1958; 1960. MR 19, 1066; MR 22 #5888.
J. N. Lyness & B. W. Ninham, "Numerical quadrature and asymptotic expansions," Math. Comp., v. 21, 1967, pp. 162-178. MR 37 #1081.
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.
Retrieve articles in Mathematics of Computation with MSC: 42.10
Retrieve articles in all journals with MSC: 42.10
Keywords: Fourier coefficients, Euler-Maclaurin summation formula, Fourier coefficient asymptotic expansion, numerical quadrature, subtracting out singularities
Article copyright: © Copyright 1971 American Mathematical Society