Let $w$ be a smooth real function of compact support. Let $f$ be a real continuous function. Then the $n$ dimensional Poisson summation formula gives us \begin{eqnarray} &&\sum_{\substack{ \mathbf{a}\in \mathbb{Z}^n} } w \left( \mathbf{a} \right) e \left( f \left(\mathbf{a} \right) \right) = \sum_{\substack{ \mathbf{k} \in \mathbb{Z}^n } } \int_{\mathbb{R}^n} w \left( \mathbf{t} \right) e \left( f \left( \mathbf{t} \right) - \mathbf{k} \cdot \mathbf{t} \right) d \mathbf{t}. \end{eqnarray}

What I was wondering was can I undo the Poisson summation on the right hand side for some of the variables? What I mean by this is the following: \begin{eqnarray} && \sum_{\substack{ \mathbf{k} \in \mathbb{Z}^n } } \int_{\mathbb{R}^n} w \left( \mathbf{t} \right) e \left( f \left( \mathbf{t} \right) - \mathbf{k} \cdot \mathbf{t} \right) d \mathbf{t} \\ &=& \sum_{\mathbf{k}' \in \mathbb{Z}^{n-1}} \int_{\mathbb{R}^{n-1}} e(- \mathbf{k}' \cdot \mathbf{t}')\left( \sum_{k_1 \in \mathbb{Z}}\int_{\mathbb{R}} w(t_1, \mathbf{t}') e(f(t_1, \mathbf{t}') - k_1 t_1) d t_1 \right) d \mathbf{t}' \\ &=& \sum_{\mathbf{k}' \in \mathbb{Z}^{n-1}} \int_{\mathbb{R}^{n-1}} e(- \mathbf{k}' \cdot \mathbf{t}')\left( \sum_{a_1 \in \mathbb{Z}} w(a_1, \mathbf{t}') e(f(a_1, \mathbf{t}') \right) d \mathbf{t}'. \end{eqnarray}

I am switching the order of summation and integral and the summation is over an infinite set, and I was wondering if this is still valid or not. Any comment would be appreciated. Thank you!

ps here $e(z) = e^{2 \pi i z}$


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