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 $e_k(x):=e^{2\pi ikx}$.

The character and finite trigonometric-polynomial clauses are choice-free. For the integral and $L^1$ clauses below, assume the Axiom of Countable Choice, as in The one-dimensional torus and its normalized Haar integral.

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.

Put $m=m_{\mathbb{T}}$. Take Borel representatives $F,G$ of the two $L^1$ classes on the Borel probability space $\mathbb{T}$ and write $\tilde{F}=F\circ q$ and $\tilde{G}=G\circ q$ for their one-periodic Borel lifts. This uses only a finite selection of representatives. The map $(x,t)\mapsto [x-t]$ is continuous on ${\mathbb{T}}^2$: the map $(x,t)\mapsto q(x-t)$ on ${\mathbb{R}}^2$ is continuous and constant on fibres of the open quotient map $q\times q$, so it descends continuously. Thus $H(x,t)=F(x-t)G(t)$ is Borel on ${\mathbb{T}}^2$.

For every nonnegative one-periodic Borel $h$, the affine reflection $t\mapsto x-t$ preserves Lebesgue measure by A linear map $T$ of ${\mathbb{R}}^n$ sends Lebesgue measurable sets to Lebesgue measurable sets, with ${\lambda }_n(T[E])=\mathrm{\mid }\det T\mathrm{\mid }{\lambda }_n(E)$ when $T$ is invertible and $T[E]$ Lebesgue null when it is not and Lebesgue outer measure, Lebesgue measurability and Lebesgue measure are unchanged by translation. It carries a half-open unit interval to another half-open unit interval; splitting the latter at its integer boundary and translating one piece by an integer, using periodicity, gives ${\int }_0^{1}h(x-t)dt={\int }_0^{1}h(u)du\mathrm{.}$ The endpoints are null. The same identity is translation invariance of the normalized torus integral. In particular, Tonelli on the finite product probability space gives ${\int }_{{\mathbb{T}}^2}\mathrm{\mid }H(x,t)\mathrm{\mid }dm(x)dm(t)={\int }_{\mathbb{T}}\mathrm{\mid }G(t)\mathrm{\mid }({\int }_{\mathbb{T}}\mathrm{\mid }F(x-t)\mathrm{\mid }dm(x))dm(t)=\mathrm{\parallel }F{\mathrm{\parallel }}_1\mathrm{\parallel }G{\mathrm{\parallel }}_1<\mathrm{\infty }\mathrm{.}$ Therefore Fubini's theorem for L^1 functions on a sigma-finite product makes $t\mapsto H(x,t)$ integrable for almost every $x$, and makes its section integral, set to zero on the measurable exceptional set, an $L^1$ function with norm at most $\mathrm{\parallel }F{\mathrm{\parallel }}_1\mathrm{\parallel }G{\mathrm{\parallel }}_1$.

If $F=F^{\mathrm{\prime }}$ and $G=G^{\mathrm{\prime }}$ almost everywhere, let $N_F,N_G$ be their Borel discrepancy sets. For $t\mathrm{\notin }{N}_G$, the $x$-section of the set where the two products differ lies in $N_F+t$ and is null by translation invariance. Tonelli applied to its indicator makes the difference set product-null; Fubini then shows that the two section integrals agree for almost every $x$. Hence the $L^1$ class is independent of representatives. If one chosen representative is bounded, the period-translation identity bounds every section integral by the product of that bound and the other factor's finite $L^1$ norm, proving the final pointwise-existence clause. Countable Choice supplies the torus integral and the Lebesgue reflection interface used above; Tonelli and Fubini then apply on the resulting probability space. No selection over all $x$ is made.