Sums and products of periodic functions

There exist two real valued periodic functions on the real line such that, for every x ∈ ℝ, f₁(x) + f₂(x) = x, but it is impossible to find two real valued periodic functions on the real line such that, for every x ∈ ℝ, f₁(x) + f₂(x) = x². The purpose of this note is to prove this result and also to study the possibility of decomposing more general polynomials into sum of periodic functions.