The Discrete Fourier Transform of (r, s)-Even Functions

Abstract

An (r, s)-even function is a special type of periodic function mod rs. These functions were defined and studied for the the first time by McCarthy. An important example for such a function is a generalization of Ramanujan sum defined by Cohen. In this paper, we give a detailed analysis of DFT of (r, s)-even functions and use it to prove some interesting results including a generalization of the Holder identity. We also use DFT to give shorter proofs of certain well known results and identities.