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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.

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    $\begingroup$ When $f$ is $1,1$-periodic and $C^3(\Bbb{R}^2)$ then yes its Fourier series converges absolutely from the relation between the Fourier coefficients of $f$ and $\partial f$ $\endgroup$ – reuns Sep 28 '19 at 14:15
  • $\begingroup$ What is $\partial f$? $\endgroup$ – Takeshi Gouda Sep 28 '19 at 14:37
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    $\begingroup$ The derivative with respect to some variable. When $g$ is $1$-periodic and $C^1$ then $\int_0^1 g'(x)e^{-2i \pi nx}dx = (2i\pi n)\int_0^1 g(x)e^{-2i \pi nx}dx$ so if $g \in C^2$ then $\int_0^1 g''(x)e^{-2i \pi nx}dx = (2i\pi n)^2\int_0^1 g(x)e^{-2i \pi nx}dx$ and hence $\sum_n |\int_0^1 g(x)e^{-2i \pi nx}dx| < \infty$. In dimension 2 it works the same way except we need to go to the 3rd derivatives $\endgroup$ – reuns Sep 28 '19 at 14:43
  • $\begingroup$ @reuns Thank you!! $\endgroup$ – Takeshi Gouda Sep 29 '19 at 14:15

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