Le capo du troupeau des s\'eries de Fourier

TL;DR

This paper explores the convergence properties of Fourier series, focusing on quadratic mean convergence for square integrable functions and Dirichlet's theorem for piecewise differentiable functions with integrable derivatives.

Contribution

It provides an exposition of key convergence theorems in Fourier series, highlighting their implications for different classes of functions.

Findings

Quadratic mean convergence for square integrable functions

Dirichlet's convergence theorem for piecewise differentiable functions

Insights into Fourier series behavior for various function classes

Abstract

A glimpse at $\sum{{\sin(kx)}\over{k}}$ gives a few lines exposition of Fourier series's quadratic mean convergence for square integrable functions and Dirichlet's convergence theorem of the Fourier serie of a piecewise differentiable function with integrable derivative.