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.

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

References in corpus (4)

• Linking the Circle and the Sieve: Ramanujan - Fourier Series
• The correlation between multiplicities of closed geodesics on the modular surface
• Ramanujan - Fourier Series and the Density of Sophie Germain Primes
• Rota meets Ramanujan: Probabilistic interpretation of Ramanujan - Fourier series

Cited by in corpus (1)

• Abel Summation of Ramanujan-Fourier Series