periodicity of an exponential sum

Question

I wish to rigorously prove that the function $f(x), x \in \mathbb{R}$ is not periodic.

A function is defined to be periodic with period $M$ if $f(x+M)=f(x), \forall x \in \mathbb{R}$.

Here $f(x) = \sum_{n=-\infty}^{\infty}e^{-(x-n)}\mu(x-n)$ where $\mu(x)=0$ iff $x\geq0$.

I have tried to prove by certain contradiction arguments but it doesn't work out. I doubt a direct proof can work here. Any help will be appreciated.

Answer 1

Assuming that the sum converges, this function is periodic, since it satisfies $f(x)=f(x+1)$. To see this, reindex the sum: \begin{eqnarray*} f(x+1)&=&\sum_{n\in\mathbb{Z}} e^{-(x+1-n)} \mu(x+1-n)\\ &=& \sum_{m\in\mathbb{Z}} e^{-(x-m)} \mu(x-m), \qquad {\rm where\ } m=n-1\\ &=& f(x). \end{eqnarray*}