KU Math 800: Lecture 2/16/2026
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Chapter 4
[theorem=%counter% (Poisson Summation Formula)] If $f \in \mathcal{F}_a$ for some $a$, then
\[\sum_{n \in \ZZ} f(n) = \sum_{n \in \ZZ}\hat{f}(n).\][/theorem]
[theorem=%counter% (Paley-Wiener Theorem (version 1); Midterm Project)] Suppose that $|\hat{f}(\xi)| \le Ae^{-2\pi a|\xi|}$ for some constants $a, A > 0$. Then $f(x)$ is the restriction to $\RR$ of a function $f(z)$ holomorphic in the strip
\[S_b = \{z \in \CC \mid |\Im(z)| < b\},\]for $0 < b < a$. [/theorem]
[theorem=%counter% (Paley-Wiener Theorem (generalized); Midterm Project)] Suppose that $f$ is continuous and of moderate ddecrease on $\RR$. Then $f$ has an extension to the complex plane that is entire with
\[|f(z)| \le Ae^{2\pi M|z|}\]for some $A > 0$, f and only if $\hat{f}$ is supported in $[-M, M]$. [/theorem]
[example=%counter%] Suppose that $f(x) = e^{-\pi x^2}$. Then we know that $\hat{f}(\xi) = f(\xi)$. That is,
\[\int_\RR e^{-\pi x^2} e^{-2\pi i x \xi} \dd{x} = e^{-\pi \xi^{2}}\]Fix $t > 0$ and $a \in \RR$. We will do a change of variables
\[x \mapsto t^{1/2} (x + a).\]Then
\[\hat{g}(t) = t^{-1/2} e^{-\pi\xi^2/t} \cdot e^{2\pi i a \xi}.\]Apply the Poisson sumation formula. Then
\[\begin{align*} \sum_{n\in \ZZ} g(n) &= \sum_{n\in \ZZ} \hat{g}(n) \end{align*}\]This implies that
\[\sum_{n = -\infty}^\infty e^{-\pi t(n + a)^2} = \sum_{n=-\infty}^\infty t^{-1/2} e^{-\pi n^2/t} \cdot e^{2\pi i n a}.\]Case 1 (used for Riemann zeta). If $a = 0$, then
\[\sum_{n=-\infty}^\infty e^{-\pi tn^2} = \sum_{n=-\infty}^\infty t^{-1/2} e^{-\pi n^2/t}.\]Set
\[\theta(t) = \sum_{n=-\infty}^\infty e^{-\pi t n^2}.\]So then $\theta(t) = t^{-1/2}\theta(1/t)$. Case 2 is $a\in \RR$ and this gives the Jacobi theta function. [/example]