A recently asked question (linked [here](https://mathoverflow.net/q/324705/9924)) deals with the remarkable identity
 $$ \sum\_{n\in\mathbb Z} \mathrm{sinc}(n+x)= \pi,\quad x\in\mathbb R, $$ 
where $\mathrm{sinc}(x)=\sin(x)/x$. 


It is easy to construct functions $f$ other than $\mathrm{sinc}(x)$ such that $\sum\_{n\in\mathbb Z} f(n+x)$ is constant for all real $x$: define $f$ outside of $[0,1)$ to ensure convergence and then let $f(x)=C-\sum\_{n\in\mathbb Z\setminus\{0\}}f(n+x)$ for $x\in[0,1)$. I wonder, however, whether there are *analytic* functions other than $\mathrm{sinc}(x)$ with this property? The set of such functions is a vector space over the complex numbers; is it finite-dimensional? If so, what is its dimension?



When is $\sum_{n\in\mathbb Z} f(x+n)$ constant?
If the Fourier transform $F(k)$ of $f(x)$ vanishes outside of the interval $(-1,1)$ then, by virtue of [Poisson summation,](https://en.wikipedia.org/wiki/Poisson_summation_formula)
$$\sum\_{n=-\infty}^\infty f(x+n)=\sum\_{n=-\infty}^\infty F(n)e^{2\pi inx}=F(0)$$ independent of $x$.


An example is $F(k)=k^2-a^2$ for $|k|<a$ and $F(k)=0$ for $|k|>a$, with $0<a<1$. Then 
$$f(x)=\frac{2}{\pi x^3}(a x \cos a x-\sin a x)\;\;\text{and}\;\;\sum\_n f(x+n)=-a^2.$$


