Period-one Fourier coefficients, partial sums, and convolution on the torus

Write $\mathbb{T}:=\mathbb{R}\mathrm{/}\mathbb{Z}$ and represent a function on $\mathbb{T}$ by a one-periodic function on $\mathbb{R}$.

For $k\in \mathbb{Z}$, define the $k$-th character $e_k(x):=e^{2\pi ikx}$.

If $f$ is integrable on one period, its Fourier coefficients are

$\widehat{f}(k):={\int }_0^{1}f(t)e^{-2\pi ikt}dt(k\in \mathbb{Z})\mathrm{.}$

For $N\ge 0$, the $N$-th Fourier partial sum of $f$ is

$S_{N}f(x):={\sum }_{\mathrm{\mid }k\mathrm{\mid }\le N}\widehat{f}(k)e_k(x)\mathrm{.}$

A trigonometric polynomial on $\mathbb{T}$ is a finite linear combination of the characters $e_k$.

If $f,g\in L^1(\mathbb{T})$, their convolution is the $L^1(\mathbb{T})$ class defined for almost every $x$ by

$(f\ast g)(x):={\int }_0^{1}f(x-t)g(t)dt,$

where $f$ and $g$ are read as one-periodic representatives. More precisely, the integrand is absolutely integrable for almost every $x$, and the displayed formula gives an almost-everywhere-defined integrable function whose class is independent of the chosen representatives. When one factor is bounded, as for the Dirichlet kernels below, the integral exists at every $x$ for which the other representative is integrable on one period.