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 this paper, $f_{T}$ 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!

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, $f_{T}$ 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!