# 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?

• You can presumably just take derivatives of that identity to produce lots of other functions (I didn't carefully check how to justify this though). – Christian Remling Mar 6 at 19:56
• Or take any translate of it. – Bleuderk Mar 7 at 8:53
• What's precisely assumed on $f$? that $f$ is analytic, and that there exists a constant $c=c(f)$ such that for every $x$, the sum $\sum_{k=-n}^nf(x+n)$ converges to $c$? it's not obvious for me that a non-absolutely convergent sum over $\mathbf{Z}$ should be evaluated only on symmetric intervals, so I'm not sure what's meant. – YCor Mar 11 at 8:16

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

• @LoïcTeyssier: Yes, the original function (Dirichlet kernel) is an example. But the Poisson summation formula comes with assumptions of course, and it would be good to state explicitly what else $f$ needs to satisfy for this to work. – Christian Remling Mar 6 at 20:05
• @ChristianRemling: yes, I realized that ;) – Loïc Teyssier Mar 6 at 20:08
• I wonder why the previous question on the same subject was removed, while this one is accepted so enthusiastically:-) – Alexandre Eremenko Mar 7 at 1:00
• If $f$ is $L^1$ then the necessary and sufficient condition is that $F(k) = 0$ for every $k \ne 0$ – reuns Mar 7 at 4:46
• To complete the answer to the question, it is easy to see that the family $(f_a)_{a>0}$, where $f_a(x)=\frac{2(ac\cos(ax)-\sin(ax)}{\pi x^3}$, spans a vector space of dimension $2^{\aleph_0}$ over $\mathbf{R}$, and hence the dimension of the considered vector space is $2^{\aleph_0}$. – YCor Mar 11 at 15:26

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.

• The OP is asking for analytic functions satisfying the given identity – Yemon Choi Mar 11 at 7:28
• Perhaps I am unaware of the definition you are using. I am using this definition en.wikipedia.org/wiki/Analytic_function and your example is not analytic in this sense at $x= \pm 1/2$ since it is not continuous on any neighbourhood of these points – Yemon Choi Mar 11 at 16:55
• @YemonChoi: It was an error. I replaced $f(x) = \text{rect}(x)$ by $f(x) = \text{rect}(x/2)$. The function $f(x) = \text{rect}(x/2)$ is analytic within $x\in[0,1)$. – u136536 Mar 11 at 18:25
• Thank you, but my interpretation of the OP's original question is that he or she wants $f$ to be analytic everywhere. Note that sinc is analytic at the origin (removable singularity) – Yemon Choi Mar 11 at 18:29
• @u136536 if you read entirely the question, you would see that your $f$ is a particular case of a more general observation by the OP. – YCor Mar 11 at 19:44