The calculation of Fourier coefficients by the Möbius inversion of the Poisson summation formula. III. Functions having algebraic singularities

Author:
J. N. Lyness

Journal:
Math. Comp. **25** (1971), 483-493

MSC:
Primary 65T05

DOI:
https://doi.org/10.1090/S0025-5718-1971-0297162-6

MathSciNet review:
0297162

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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 where being analytic and . 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 and at , together with trapezoidal rule sums over [0, 1] of . Some of the incidental constants are values of the generalized zeta function .

**[1]**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**[2]**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**0293846**, https://doi.org/10.1090/S0025-5718-1971-0293846-4**[3]**A. Erdélyi,*Asymptotic representations of Fourier integrals and the method of stationary phase*, J. Soc. Indust. Appl. Math.**3**(1955), 17–27. MR**0070744****[4]**J. N. Lyness,*Adjusted forms of the Fourier coefficient asymptotic expansion and applications in numerical quadrature*, Math. Comp.**25**(1971), 87–104. MR**0290020**, https://doi.org/10.1090/S0025-5718-1971-0290020-2**[5]**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

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1971-0297162-6

Keywords:
Fourier coefficients,
Euler-Maclaurin summation formula,
Fourier coefficient asymptotic expansion,
trigonometric integration,
numerical quadrature,
ignoring the singularity

Article copyright:
© Copyright 1971
American Mathematical Society