The spectral decomposition of $|\theta|^2$
2016
Let $\theta$ be an elementary theta function, such as the classical Jacobi theta function. We establish a spectral decomposition and surprisingly strong asymptotic formulas for $\langle |\theta|^2, \varphi \rangle$ as $\varphi$ traverses a sequence of Hecke-translates of a nice enough fixed function. The subtlety is that typically $|\theta|^2 \notin L^2$.
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