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Question Number 120528 by snipers237 last updated on 01/Nov/20

Let  a>0 , g(x)=Σ_(n∈Z) exp(−(((x−n)^2 )/a))  Find the Fourier coefficients of   g  and deduce that for x∈R^∗    Σ_(n∈Z) exp(−((πn^2 )/x^2 ))= x .Σ_(n∈Z) exp(−πn^2 x^2 )

$${Let}\:\:{a}>\mathrm{0}\:,\:{g}\left({x}\right)=\underset{{n}\in\mathbb{Z}} {\sum}{exp}\left(−\frac{\left({x}−{n}\right)^{\mathrm{2}} }{{a}}\right) \\ $$ $${Find}\:{the}\:{Fourier}\:{coefficients}\:{of}\:\:\:{g} \\ $$ $${and}\:{deduce}\:{that}\:{for}\:{x}\in\mathbb{R}^{\ast} \\ $$ $$\:\underset{{n}\in\mathbb{Z}} {\sum}{exp}\left(−\frac{\pi{n}^{\mathrm{2}} }{{x}^{\mathrm{2}} }\right)=\:{x}\:.\underset{{n}\in\mathbb{Z}} {\sum}{exp}\left(−\pi{n}^{\mathrm{2}} {x}^{\mathrm{2}} \right) \\ $$ $$ \\ $$

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