When is $\sum_{n\in\mathbb Z} f(x+n)$ constant? A recently asked question (linked here) 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?
 A: If the Fourier transform $F(k)$ of $f(x)$ vanishes outside of the interval $(-1,1)$ then, by virtue of Poisson summation,
$$\sum_{n=-\infty}^\infty f(x+n)=\sum_{n=-\infty}^\infty F(n)e^{2\pi inx}=F(0)$$ 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 
$$f(x)=\frac{2}{\pi x^3}(a x \cos a x-\sin a x)\;\;\text{and}\;\;\sum_n f(x+n)=-a^2.$$
A: A simplest case is a rectangular function defined as
$${\text{rect}}\left( x \right) = \left\{ \begin{aligned}
1, & \quad  - 1/2 < x < 1/2\hfill \\
1/2, & \quad \left| x \right| = 1/2 \hfill \\
0, & \quad {\text{otherwise}}{\text{.}} \hfill \\ 
\end{aligned}  \right.$$
We can see that if $f(x) = \text{rect}(x/2)$, then:
$$\sum_{n=-\infty}^\infty \text{rect}(x/2+n)=1$$
and the requirement:
$$f(x)=C-\sum_{n\in\mathbb Z\setminus\{0\}}f(n+x), \qquad x\in[0,1)$$
is satisfied. The function $f(x) = \text{rect}(x/2)$ is analytic when $x\in[0,1)$.
The rectangular function is the Fourier transform of the sinc function.
