Convergence of Fourier series

Definition

Let $f \colon \R \to \C$ be $2\pi$-periodic and integrable on $[-\pi, \pi]$, and let $S_N f(x) = \sum_{n=-N}^{N} \hat f(n) e^{inx}$ be the symmetric partial sums of its Fourier series.

1. The Fourier series of $f$ converges at $x \in \R$ if the limit $\lim_{N \to \infty} S_N f(x)$ exists in $\C$. It converges to $f$ at $x$ if this limit equals $f(x)$.
2. It converges pointwise on a set $E \subseteq \R$ if it converges at each $x \in E$ (see pointwise convergence). It converges almost everywhere if it converges to $f$ at each $x$ outside a set of measure zero.
3. It converges uniformly to $f$ on $E$ if $\sup_{x \in E} \abs{S_N f(x) - f(x)} \to 0$ as $N \to \infty$ (see uniform convergence).
4. For $1 \le p < \infty$ and $f \in L^p(\mathbb{T})$, it converges in $L^p$ if $\int_{-\pi}^{\pi} \abs{S_N f - f}^p \, dx \to 0$ as $N \to \infty$, in the Lebesgue space $L^p(\mathbb{T})$.