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For a $2\pi$ periodic distribution $u\in {\mathcal{S}}^{\prime }(\mathbb{R})$, choose a smooth periodic partition of unity $\Psi$, and define $c_n=(2\pi {)}^{-1}⟨u,\Psi e^{-inx}⟩$. A finite Schwartz seminorm bound on the fixed compact support gives $|c_n|\le C(1+|n|{)}^N$. For a Schwartz function $\varphi$, the Poisson summation formula and Fourier inversion imply $P\hat{\varphi }(x)={\sum }_n\varphi (-n)e^{inx}$. Localizing the pairing for $u$ givesproving the displayed Fourier transform formula. The series converges in tempered distributions because its coefficients have polynomial growth while $\varphi (n)$ decreases rapidly. Applying the inverse Fourier transform gives $u={\sum }_n{c}_n{e}^{inx}$ in tempered distributions. These are the smooth-window Fourier coefficients of a periodic distribution, and the transform formula shows independence of $\Psi$.

A distribution $u$ is a $2\pi$ periodic distribution whenIteration gives invariance under all integer multiples of $2\pi$. For a tempered distribution the identity also holds for every Schwartz function, since test functions are dense in the Schwartz space: multiplying by cutoff functions $\chi (x/R)$ equal to one near zero gives such an approximation in every Schwartz seminorm.

For $\varphi \in \mathcal{S}(\mathbb{R})$ define $a_k(x)=\Psi (x+2\pi k)\varphi (x)$. The smooth periodic partition of unity gives $\varphi ={\sum }_k{a}_k$ with convergence in the Schwartz space. Indeed, each $a_k$ is supported where $|x+2\pi k|\le 2\pi$, and the generalized Leibniz rule and arbitrarily rapid decay of the derivatives of $\varphi$ give, for every $m,n,M$,Consequently continuity and periodicity of the tempered distribution giveFor the last interchange, all summands have the fixed compact support of $\Psi$ and their derivatives decrease faster than any power of $|k|$. Hence the series converges in the space of test functions, as well as in the Schwartz space. Notice that $P\varphi$ itself need not be a Schwartz function; the factor $\Psi$ makes the pairing legitimate.

The $2\pi$ periodization of a Schwartz function isFor every nonnegative integer $j$, the Schwartz seminorms give $|{\varphi }^{(j)}(x+2\pi k)|\le C_j(1+|k|{)}^{-2}$ uniformly for $x\in [0,2\pi ]$. The Weierstrass M-test gives absolute and uniform convergence there of the defining series and all its differentiated series. Termwise differentiation proves that $P\varphi$ is a smooth function, with $(P\varphi {)}^{(j)}=P({\varphi }^{(j)})$. Reindexing the absolutely convergent series proves $P\varphi (x+2\pi )=P\varphi (x)$, so it is a periodic function.

For an explicit smooth periodic partition of unity, choose the bump functionIt is a test function, positive on $[-\pi ,\pi ]$. The smooth function $S(x)=P\eta (x)$ is $2\pi$ periodic and strictly positive, since every $x$ has a translate in $[-\pi ,\pi ]$. SetDivision preserves smoothness because $S>0$, and the support remains contained in $[-2\pi ,2\pi ]$.

Use the smooth periodic partition of unity $\Psi$ above and define the Fourier coefficients of the periodic distribution byThese are the smooth-window Fourier coefficients of a periodic distribution for period $2\pi$. The finite Schwartz seminorm bound for $u$, restricted to the fixed compact support of $\Psi$, gives $|c_n|\le C(1+|n|{)}^N$ for some $C,N$. Indeed, only finitely many derivatives of $\Psi e^{-inx}$ occur, and each contributes at most a fixed power of $|n|$.

For a Schwartz function $\varphi$, the preceding Fourier series formula and $\hat{\hat{\varphi }}(n)=2\pi \varphi (-n)$ yieldAll derivatives of this series converge uniformly. The localization formula in the previous answer therefore givesConsequently the Fourier transform of a periodic distribution isThe polynomial bound on $c_n$ and rapid decay of $\varphi (n)$ make every displayed pairing absolutely convergent. In particular there are no derivatives of Dirac delta distributions in this formula.

The inverse Fourier transform of ${\delta }_n$ is $(2\pi {)}^{-1}{e}^{inx}$. Applying continuous Fourier inversion on tempered distributions to the symmetric partial sums givesExplicitly, for every Schwartz function $\theta$, the pairing of the partial sum with $\theta$ is ${\sum }_{|n|\le M}{c}_n\hat{\theta }(-n)$, which converges absolutely to $⟨u,\theta ⟩$. The Fourier transform formula also proves that $c_n$ is independent of the chosen smooth periodic partition of unity. For a locally integrable $2\pi$ periodic function it reduces, by splitting the integral into periods, to $c_n=(2\pi {)}^{-1}{\int }_0^{2\pi }u(x)e^{-inx}dx$.