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The discrete Fourier transform of \(r\)-even functions. (English) Zbl 1262.11005

A function \(f: \mathbb N\to\mathbb C\) is \(r\)-even, if \(f((n,r))= f(n)\) for all \(n\in\mathbb N\). The discrete Fourier transform of such a function is \[ \widehat f(n)= \sum_{k\bmod r} f(k)\exp(-2\pi ikn/r). \] It is again \(r\)-even. The authors study this transformation and gain direct proofs of known properties and results of even functions such as the Ramanujan sums.

MSC:

11A25 Arithmetic functions; related numbers; inversion formulas
11L03 Trigonometric and exponential sums (general theory)
11N37 Asymptotic results on arithmetic functions
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