Let $\Lambda \subset \Lambda' \subset \mathbb{R}^n$ be lattices. Let $f : \Lambda' \rightarrow \mathcal{H}$ be a $a$-Lipschitz function, where $\mathcal{H}$ is a finite-dimensional Hilbert Space. Assume that $f$ is $\Lambda$-periodic. Kirszbraun theorem states that there exists a function $f' : \mathbb{R}^n \rightarrow \mathcal{H}$ such that $f'|_{\Lambda'} = f$ and $f'$ is $a$-Lipschitz.
My question is the following: can $f$ be extended in a Lipschitz way to a $\Lambda$-periodic function?
For the case $n=1$, $\Lambda = ab\mathbb{Z}, \Lambda' = b\mathbb{Z}$ we can use the formula $f'(x) = f(b \cdot \lfloor x/b\rfloor)\cdot (1-\lbrace x/b\rbrace) + f(b \cdot \lfloor x/b\rfloor+1)\cdot \lbrace x/b\rbrace$, which works, but I do not know how to generalize it to higher dimensions.
Is it something possible ? Has it been done before ?
Thank you in advance !
