Let $f: \mathbb{R}^n \to \mathbb{R}^n$ such that
- $f$ is invariant w.r.t. addition of scalars: $\forall x \in \mathbb{R}^n, \lambda \in \mathbb{R}: f(x + \lambda) = f(x)$, and
- $f$ is periodic in each coordinate with period $L > 0$: For $i = 1, \dots, n$ and $x \in \mathbb{R}^n$: $f(x_1, \dots, x_{i-1}, x_i + L, x_{i+1}, \dots, x_n) = f(x)$.
I've been trying to find a simple representation of such functions, but have been stuck for a bit now. I've done the following so far:
Let $\mathbb{1} \in \mathbb{R}^n$ be the constant vector of ones. Clearly, $f$ is uniquely determined by its values on $\mathbb{1}^\perp$, as $f(x) = f(x - \bar{x})$ and $\mathbb{1}^\top (x - \bar{x}) = \sum_{i=1}^n (x_i - \bar{x}) = 0$. Combined with periodicity in each coordinate, $f$ is uniquely determined by its values on the intersection of some $n$-dimensional cube (say e.g. $[-1, 0] \times [0, 1]^{n-1}$) and $\mathbb{1}^\perp$. However, the values on this intersection still cannot be chosen freely as there are some constraints left by combining 1. and 2.: For any $i =1, \dots, n$: $$f(x - \bar{x}) = f\left(x_1 - \bar{x} - \frac{L}{n}, \dots, x_i - \bar{x} + \frac{(n-1)L}{n}, \dots, x_n - \bar{x} - \frac{L}{n}\right).$$
For $n=2$, these constraints should simply boil down to $f$ being $L/2$-periodic in each coordinate on $\mathbb{1}^\perp$, so - if I'm not mistaken - any such $f$ can precisely be written as $$f(x) = g\left( (x - \bar{x})\text{ mod }\frac{L}{2}\right), \qquad (\ast)$$ where $g: [0, L/2)^2 \to \mathbb{R}^2$ can be chosen arbitrarily (Edit: $g$ permutation equivariant).
I was wondering if this generalizes to $n > 2$ in some way, i.e. if one can find anything reasonably simple that is similar to $(\ast)$. Any help, also a negative answer, would be very much appreciated:)
Edit: I forgot to mention that I also need
- $f$ is permutation equivariant, i.e. for all permutation matrices $P \in \mathbb{R}^{n \times n}$: $f(Px) = Pf(x)$.