The calculation of Fourier coefficients by the Möbius inversion of the Poisson summation formula. I. Functions whose early derivatives are continuous
TLDR
In this paper, the M6bius inversion technique is applied to the Poisson summation formula, which results in expressions for the remainder term in the Fourier coefficient asymptotic expansion as an infinite series.Abstract:
The M6bius inversion technique is applied to the Poisson summation formula. This results in expressions for the remainder term in the Fourier coefficient asymptotic expansion as an infinite series. Each element of this series is a remainder term in the corresponding Euler-Maclaurin summation formula, and the series has specified convergence properties. These expressions may be used as the basis for the numerical evaluation of sets of Fourier coefficients. The organization of such a calculation is described, and discussed in the context of a broad comparison between this approach and various other standard methods.read more
Citations
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A numerical method for the integration of oscillatory functions
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An algorithm based on the FFT for a generalized Chebyshev interpolation
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Adjusted forms of the Fourier coefficient asymptotic expansion and applications in numerical quadrature
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References
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An algorithm for the machine calculation of complex Fourier series
J.W. Cooley,John W. Tukey +1 more
TL;DR: Good generalized these methods and gave elegant algorithms for which one class of applications is the calculation of Fourier series, applicable to certain problems in which one must multiply an N-vector by an N X N matrix which can be factored into m sparse matrices.
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An Introduction to the Theory of Numbers
TL;DR: The fifth edition of the introduction to the theory of numbers has been published by as discussed by the authors, and the main changes are in the notes at the end of each chapter, where the author seeks to provide up-to-date references for the reader who wishes to pursue a particular topic further and to present a reasonably accurate account of the present state of knowledge.
Book
Numerical Methods for Scientists and Engineers
TL;DR: This inexpensive paperback edition of a groundbreaking classic is unique in its emphasis on the frequency approach and its use in the solution of problems.