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If you are given a smooth function, how to verify whether its priodization is well-defined and also smooth (or not)?

For instance, suppose, there is a series $$ f(x) = \sum_{n=-\infty}^{\infty}e^{-\tau (n+x)^2}, \quad \tau >0 $$ then, what is the strategy for proving it is well-defined and $C^\infty(\mathbb R)$? Clearly, this is simply a periodization of the following function (which is, for the sake of simplicity, taken to be an element of the Schwartz space $S(\mathbb R)$): $$ \tilde f(x) = e^{-\tau x^2} $$ with period 1: $$ f(x) = \sum_{n=-\infty}^\infty \tilde f(x+n) $$

Thank you very much for any hint!

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  • $\begingroup$ It suffices to show that the derivatives of $\tilde{f}$ decay sufficiently fast at $\pm \infty$, so that the series $\sum_n \tilde{f}^{(k)}(x + n)$ converges locally uniformly. Do you have some irregular example of $\tilde{f}$ in mind? $\endgroup$
    – Mateusz Kwaśnicki
    Commented Mar 29, 2019 at 12:47
  • $\begingroup$ @MateuszKwaśnicki thank you for your comment! Probably, I formulated the question not generally enough: I should have asked about whether $\tilde f\in C^\infty$ implies $f\in C^\infty$, and about a possible strategy of proving/disproving this implication in the general case. Nevertheless, I appreciate your help: a strategy you've pointed out seems to be the strategy. Thanks a lot! $\endgroup$
    – jonathan wolf
    Commented Mar 29, 2019 at 13:30
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    $\begingroup$ No, $f$ need not be smooth, in fact, not even continuous. Take $\tilde f$ to be the sum of increasingly thin bumps located at $n+1/n$: then $f$ will fail to be continuous at $0$. $\endgroup$
    – Mateusz Kwaśnicki
    Commented Mar 29, 2019 at 16:18

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