A summability method for the arithmetic Fourier transform (Q1335004)

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.