The Discrete Fourier Transform of $\mathbf{(r,s)-}$even functions

TL;DR

This paper analyzes the discrete Fourier transform of (r,s)-even functions, a class of periodic functions, and uses it to generalize identities and provide shorter proofs of known results.

Contribution

It offers a detailed analysis of the DFT of (r,s)-even functions and generalizes the Hölder identity, enhancing understanding of these functions and their properties.

Findings

Generalization of the Hölder identity.

Shorter proofs of known identities.

Deeper understanding of DFT of (r,s)-even functions.

Abstract

An $(r,s)$-even function is a special type of periodic function mod $r^s$. 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 H\"{o}lder identity. We also use DFT to give shorter proofs of certain well known results and identities .