The calculation of Fourier coefficients by the Möbius inversion of the Poisson summation formula. III. Functions having algebraic singularities
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- by J. N. Lyness PDF
- Math. Comp. 25 (1971), 483-493 Request permission
Abstract:
The purpose of this paper is to extend the MIPS theory described in Parts I and II to functions having algebraic singularities. As in the simpler cases, the theory is based on expressing the remainder term in the appropriate Fourier coefficient asymptotic expansion as an infinite series, each element of which is a remainder in the Euler-Maclaurin summation formula. In this way, an expression is found for $\smallint _a^bf(x)\cos 2\pi mx\;dx\;(0 \leqq a < b \leqq 1)$ where $f(x) = {(x - a)^\alpha }{(b - x)^\beta }\phi (x),\phi (x)$ being analytic and $\alpha ,\beta > - 1$. This expression, and variants of it form a convenient basis for the numerical calculation of a set of Fourier coefficients. The calculation requires approximate values of the first few derivatives of $\phi (x)\;{\text {at}}\;x = a$ and at $x = b$, together with trapezoidal rule sums over [0, 1] of $f(x)$. Some of the incidental constants are values of the generalized zeta function $\zeta (s,a)$.References
- 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
- J. N. Lyness, The calculation of Fourier coefficients by the Möbius inversion of the Poisson summation formula. II. Piecewise continuous functions and functions with poles near the interval $[0,\,1]$, Math. Comp. 25 (1971), 59–78. MR 293846, DOI 10.1090/S0025-5718-1971-0293846-4
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- J. N. Lyness, Adjusted forms of the Fourier coefficient asymptotic expansion and applications in numerical quadrature, Math. Comp. 25 (1971), 87–104. MR 290020, DOI 10.1090/S0025-5718-1971-0290020-2
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Additional Information
- © Copyright 1971 American Mathematical Society
- Journal: Math. Comp. 25 (1971), 483-493
- MSC: Primary 65T05
- DOI: https://doi.org/10.1090/S0025-5718-1971-0297162-6
- MathSciNet review: 0297162