Convolution and Cross-Correlation of Ramanujan-Fourier Series

TL;DR

This paper explores the relationship between almost periodic functions and Ramanujan-Fourier series, establishing connections for convolution and cross-correlation without requiring uniform convergence, and extends the Wiener-Khinchin formula to arithmetic functions.

Contribution

It introduces a novel approach using almost periodic functions to analyze Ramanujan-Fourier series and extends classical results like Wiener-Khinchin to this context.

Findings

Established connection between almost periodic functions and Ramanujan-Fourier series.

Proved convolution and cross-correlation properties without uniform convergence.

Extended Wiener-Khinchin formula to arithmetic functions with 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.