This question is motivated by my previous [post](https://math.stackexchange.com/questions/13344/proof-for-an-integral-involving-sinc-function) in SE (math.stackexchange.com).


Prove or disprove that $\frac{\sin x}{x}$ is the only nonzero entire (i.e. analytic everywhere), or continuous,
function, $f(x)$ on $\mathbb{R}$ such that $$\int\_{-\infty}^\infty f(x) dx=\int\_{-\infty}^\infty f(x)^2 dx=\sum\_{-\infty}^\infty f(n) =\sum\_{-\infty}^\infty f(n)^2 $$



Continuous or analytic functions with this property of sinc function
Seems like the Poisson formula is well-forgotten nowdays. Assume that $f$ is a real valued even Schwartz function and pass to its Fourier transform $g$ (with $2\pi$ in the exponent). Then the conditions are $$g(0)=\int\_{\mathbb R} g^2=\sum\_{k\in\mathbb Z} g(k)=\int\_{[-1/2, 1/2]}G^2$$ where $G(x)=\sum\_{k\in\mathbb Z} g(x+k)$ is the $1$-periodization of $g$. Not to bother about $1$-periodization too much, assume that $g$ is supported on $[-0.3,0.3]$ (that'll take care of analyticity of $f$ too). Then we just need a smooth real-valued even function on that interval whose value at $0$ equals the integral of its square. They are plenty.


This would be a nice question for an analysis qualifier exam. :) I'll try to propose it for the next one and see if it flies.


