Is there any known characterization of the functions $\mathbb{R \to R}$ that can be written as a sum of (a finite family of) periodic functions?

It is well-known that the identity function on $\mathbb{R}$ is a sum of two periodic functions (proof: decompose $\mathbb{R}$ into a direct sum of two vector spaces over $\mathbb{Q}$); similarly every polynomial function of degree $d$ is a sum of $d+1$ periodic functions. Such functions are also dense for the topology of pointwise convergence according to https://doi.org/10.1515/gmj-2019-2076 but I couldn't find a characterization in the literature.