Let $a₁, ..., a_k$ be arbitrary nonzero real numbers. An $(a₁, ..., a_k)$-decomposition of a function f:ℝ → ℝ is a sum $f₁ + ⋯ + f_k = f$ where $f_i: ℝ → ℝ$ is an $a_i$-periodic function. Such a decomposition is not unique because there are several solutions of the equation $h₁ + ⋯ + h_k = 0$ with $h_i : ℝ → ℝ a_i$-periodic. We will give solutions of this equation with a certain simple structure (trivial solutions) and study whether there exist other solutions or not. If not, we say that the $(a₁, ..., a_k)$-decomposition is essentially unique. We characterize those periods for which essential uniqueness holds.