Convolution and Cross-Correlation of Ramanujan-Fourier Series
Abstract
This paper uses the machinery of almost periodic functions to prove that even without uniform convergence the connection between a pair of almost periodic functions and the constants of the associated Fourier series exists for both the convolution and cross-correlation. The general results for two almost periodic functions are narrowed and applied to Ramanujan sums and finally applied to support the specific relation of the Wiener-Khinchin formula for arithemic functions with a Ramanujan-Fourier Series.
Cite
@article{arxiv.0805.0284,
title = {Convolution and Cross-Correlation of Ramanujan-Fourier Series},
author = {John Washburn},
journal= {arXiv preprint arXiv:0805.0284},
year = {2015}
}
Comments
The paper fails to establish the continuity of the a.p. functions involved required to uniformly approximate the a.p. function with the Bochner-Fejer polynomials. Without this uniform convergence the key interchange of limits is not justified