Consider a function $f\in\mathcal{C}_0(\mathbb{R})$, where $\mathcal{C}_0(\mathbb{R})$ denotes the space of *bounded continuous functions vanishing at infinity*. I am interested in the *$T$-periodisation* of such a function, defined as:
$$f_{T}(t)=\sum_{n\in\mathbb{Z}} f(t-nT),\quad \forall t\in \mathbb{R}.$$
As explained in [Fischer - On the duality of discrete and periodic functions][1], $f_{T}$ is a $T$-periodic *tempered distribution* if $f$ is a *rapidly decaying function* —i.e. vanishing at infinity faster than any polynomial. 

My question concerns the regularity of $f_T$:

**For which functions $f\in\mathcal{C}_0(\mathbb{R})$ is the periodised generalised function $f_{T}$ defined above an *ordinary, continuous function*?** 

In other words, what should be the assumptions on $f$ so that its periodisation is 
continuous?  

Any lead would be greatly appreciated. Thank you very much in advance!

  [1]: https://www.mdpi.com/2227-7390/3/2/299