# A discrete approach to Wirtinger's inequality

## TL;DR

This paper develops a discrete version of Wirtinger's inequality for piecewise functions, offering a new elementary proof and insights into the equality case, while connecting to Fourier series development.

## Contribution

It introduces a novel discrete approach to Wirtinger's inequality and provides a new elementary proof, also exploring the equality case and Fourier series implications.

## Key findings

- Discrete version of Wirtinger's inequality derived
- Elementary proof of the inequality established
- Connection to Fourier series development of periodic functions

## Abstract

Considering Wirtinger's inequality for piece-wise equipartite functions we find a discrete version of this classical inequality. The main tool we use is the theorem of classification of isometries. Our approach provides a new elementary proof of Wirtinger's inequality that also allows to study the case of equality. Moreover it leads in a natural way to the Fourier series development of $2\pi$-periodic functions.