= Periodization of a Schwartz function
{title2=$P_f(x)=\sum_{n\in\mathbb Z}f(x+n)$}
For a <Schwartz function> on the <real numbers>, its periodization is a smooth <periodic function> of period one. Rapid decay gives <uniform convergence> on $[0,1]$ of the series and each differentiated series. Under the <Fourier transform> convention $\widehat f(\xi)=\int f(x)e^{-2\pi ix\xi}\,dx$, its $m$th <Fourier coefficient> is $\widehat f(m)$: integrate termwise over one period and combine the translated intervals. The rapidly decreasing coefficients give a <Fourier series> with <uniform convergence>. Evaluation at zero proves the <Poisson summation formula>.
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