Theorem 1 of this paper shows that
For every positve integer $K$ and for every nonempty bounded domain $\mathcal X \subseteq \mathbb R^d$, the restriction on $\mathcal X$ of any function $h: \mathbb R^d \rightarrow \mathbb R^d$ satisfying:
- Invertibility: $h$ is invertible.
Lipschitz moothness: There exists $L \in [0,\infty)$ such that $\|h'(x)u-h'(y)u\| \le L\|x-y\|\|u\|$, for all $x, u \in \mathbb R^d$.
Positive orientation: $\det(h'(x_0)) > 0$ for some $x_0 \in \mathcal X$.
Lipschitz inverse: $\|h^{-1}\|_{\text{Lip}} < \infty$,
can be represented as a composition $h|_{\mathcal X} = h_1\circ h_2 \circ \ldots \circ h_K$ of near-identity functions, in the sense that $$\||h_k - \operatorname{Id}_d\|_{\text{Lip}} = \mathcal O (\log(K)/K),\text{ for all }1 \le k \le K. $$
Unfortunately, the provided proof is very "manual".
Question
- Is there a result from functional analysis, approximation theory, or a simple argument, which would automatically guarantee the existence of such an approximation in the above theorem or a similar statement ?
