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My question is a follow-up to Abdelmalek Abdesselam's recent post What makes Gaussian distributions special? Local field version? asking about various characterizations of (real-valued) Gaussian ...

I recently meet the following two weird "Fourier transform" questions. (I), Suppose that $F$ is a $p$-adic field (the same question can be asked over any local field, including $\mathbb{R}$ ...