A summability method for the arithmetic Fourier transform
type: Item
instance of: scholarly article
title: A summability method for the arithmetic Fourier transform
zbMATH Open document ID: 0821.65093
DOI: 10.1007/BF01955877
author: W. J. Walker
published in: BIT
publication date: 1995-09-27T00:00:00Z
review text: The arithmetic Fourier transform can be used to calculate the Fourier cosine coefficients \(a_ n\) of an even \(2 \pi\)-periodic function \(f\). Assume that \(f\) is sufficiently smooth, and that \(a_ 0 = 0\). If \(\mu\) denotes the Möbius function and if \(S(n) : = {1 \over n} \sum^{n - 1}_{j = 0} f({2 \pi j \over n})\), then \(a_ n = \sum^ \infty_{k = 1} \mu (k) S(nk)\), \(n \geq 1\). The author introduces a modified version of this method and gives a numerical example.
reviewed by: Manfred Tasche
Mathematics Subject Classification ID: 65T40
Mathematics Subject Classification ID: 42A16
zbMATH DE Number: 644797
zbMATH Keywords: summability method
zbMATH Keywords: periodic function
zbMATH Keywords: arithmetic Fourier transform
zbMATH Keywords: Fourier cosine coefficients
zbMATH Keywords: Möbius function
zbMATH Keywords: numerical example
MaRDI profile type: Publication
cites work: Representation of Fourier Integrals as Sums. III
cites work: Q5556050
cites work: Fourier analysis and signal processing by use of the Mobius inversion formula
cites work: A Sampling Theorem for Analytic Functions
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cites work: On implementing the arithmetic Fourier transform
cites work: Q5834958
full work available at URL: https://doi.org/10.1007/bf01955877
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description: scientific article; zbMATH DE number 644797