= 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 $P_f$ is a <smooth> <periodic function> of <period of a function> one. Rapid decay gives <uniform convergence> on $[0,1]$ of the defining <series> and each differentiated <series>. Its $k$th <Fourier coefficient> is
$$
c_k=\int_0^1P_f(x)e^{-2\pi ikx}\,dx
=\sum_{n\in\mathbb Z}\int_n^{n+1}f(u)e^{-2\pi iku}\,du
=\widehat f(k).
$$
The change of variable $u=x+n$ uses $e^{2\pi ikn}=1$. Rapidly decreasing <Fourier coefficients> give the absolutely and uniformly convergent <Fourier series> $P_f(x)=\sum_k\widehat f(k)e^{2\pi ikx}$. Evaluation at $x=0$ proves the <Poisson summation formula>.<Number Theory I, MIT 18.785, Fall 2021/Lecture 17: The functional equation>[Lecture 17, §17.1, Theorem 17.8 and proof, printed p. 3: periodization and Fourier coefficients]{ref}
The same identity explains <aliasing> in the <Nyquist–Shannon sampling theorem>: applying the <Poisson summation formula> to the <Schwartz function> $x\mapsto f(x)e^{-2\pi i\xi x}$ gives
$$
\sum_{n\in\mathbb Z}f(n)e^{-2\pi i\xi n}
=\sum_{k\in\mathbb Z}\widehat f(\xi+k).
$$
The sampled spectrum is periodic in <frequency>, and shifted copies of the continuous <Fourier transform> add together. When those copies overlap, the samples cannot separate their contributions. The <Nyquist–Shannon sampling theorem> adds a <bandlimited function> hypothesis to make reconstruction possible.
Back to article page
