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Let $f=\sum_{n=1}^N k_n I_{E_n}$ where $E_n$ are Borel subsets of $\mathbb{R}^n$ and $k_n\in \mathbb{R}^m$ with non-negative entries, and let $\mu$ be a finite Borel measure on $\mathbb{R}^n$. What is the best rate of approximation, in $L^1(\mathbb{R}^n,\mathbb{R}^m)$ of $f$ by Lipschitz functions? That is, given any positive integer $n$ and some $L>0$, what order is: $$ \inf_{g \in Lip(\mathbb{R}^n,\mathbb{R}^m); Lip(g)\leq L}\,\int_{x\in \mathbb{R}^n} \|g(x)-f(x)\|d\mu(x)? $$

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    $\begingroup$ The infimum is 0 because the Lip functions are dense in $L^1$. Maybe you mean the best approximation by Lipschitz functions with a given Lipschitz constant? $\endgroup$ – Pietro Majer Mar 16 at 18:15
  • $\begingroup$ Right good point; I updated the post. $\endgroup$ – Zorn's Llama Mar 17 at 8:47
  • $\begingroup$ I guess, even in general for $f\in L^1(\mathbb{R}^n,\mathbb{R}^m)$, it may be given in terms of the "modulus of $L^1$-continuity" of $f$, that is $\omega(\delta):=\sup_{\|h\|\le\delta}\|f-f_h\|_1,$ where $f_h(x):=f(x-h)$. $\endgroup$ – Pietro Majer Mar 17 at 15:41
  • $\begingroup$ @Pietro I also figured something like this; but wouldn't we need $f$ to belong to a Sobolev class? $\endgroup$ – Zorn's Llama Mar 17 at 18:09
  • $\begingroup$ I don't think so; the problem you posed is meaningful even for $f\in L^1(\mathbb{R}^n,\mathbb{R}^m)$, that is to estimate the point-set distance $$d(k):=\rm{dist}\big(f, \rm{Lip_k}(\mathbb{R}^n,\mathbb{R}^m)\big),$$ and its decay for $k\to+\infty$. $\endgroup$ – Pietro Majer Mar 17 at 18:47

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