Fourier series — coefficients, definition, Dirichlet theorem

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\title{Fourier Series --- Definitions and Convergence}
\author{}
\date{}

\begin{document}
\maketitle

\section{Fourier coefficients}

\begin{definition}{Fourier coefficients}{coef-fourier}
Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a $2\pi$-periodic continuous function. The \textbf{Fourier coefficients} of $f$ are the numbers defined for $n \in \mathbb{N}^*$ by:
\[
  a_n = \frac{1}{\pi} \int_{0}^{2\pi} f(x) \cos(nx) \, \mathrm{d}x,
  \qquad
  b_n = \frac{1}{\pi} \int_{0}^{2\pi} f(x) \sin(nx) \, \mathrm{d}x.
\]
We additionally set:
\[
  a_0 = \frac{1}{2\pi} \int_{0}^{2\pi} f(x) \, \mathrm{d}x \qquad \text{and} \qquad b_0 = 0.
\]
\end{definition}

\begin{property}{Parity}{parity}
Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be $2\pi$-periodic and continuous.
\begin{itemize}[nosep]
  \item If $f$ is \emph{even}, then $b_n = 0$ for every $n \in \mathbb{N}$.
  \item If $f$ is \emph{odd}, then $a_n = 0$ for every $n \in \mathbb{N}$.
\end{itemize}
\end{property}

\section{Fourier series}

\begin{definition}{Fourier series}{serie-fourier}
Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be $2\pi$-periodic and continuous. The \textbf{Fourier series} of $f$ is the series with general term:
\[
  a_n \cos(nx) + b_n \sin(nx),
\]
where $a_n$ and $b_n$ are the Fourier coefficients of $f$.
\end{definition}

\begin{remark}
The Fourier series of $f$ is then written:
\[
  \sum_{n \geq 0} \big( a_n \cos(nx) + b_n \sin(nx) \big).
\]
\end{remark}

\section{Convergence}

\begin{theorem}{Dirichlet's theorem}{dirichlet}
Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be $2\pi$-periodic and differentiable. Then the Fourier series of $f$ converges at every point $x \in \mathbb{R}$, and:
\[
  f(x) = a_0 + \sum_{n=1}^{+\infty} \big( a_n \cos(nx) + b_n \sin(nx) \big).
\]
\end{theorem}

\begin{remark}
Under weaker assumptions (e.g.\ $f$ piecewise continuous and piecewise $\mathcal{C}^1$), the Fourier series converges to $\dfrac{f(x^+) + f(x^-)}{2}$ at every point. This is the most general form of Dirichlet's theorem.
\end{remark}

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