I have a smooth function compact support $f(x,y)$. Then the Poisson summation formula gives $$ \sum_{n_1, n_2 \in \mathbb{Z}} f(n_1, n_2) = \sum_{m_1, m_2} \int_{\mathbb{R}^2} f(z_1,z_2) e^{- 2 \pi i (m_1 z_1 + m_2 z_2) } dz_1dz_2. $$ I know that the sum on the right hand side is convergent, but is it always absolutely convergent also? And how can one show that? Thank you.
2026-02-22 23:36:14.1771803374
Question on Poisson summation formula and Fourier transform
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