# Best approximation of piecewise constant function by Lipschitz functions

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: $$\inf_{g \in Lip(\mathbb{R}^n,\mathbb{R}^m); Lip(g)\leq L}\,\int_{x\in \mathbb{R}^n} \|g(x)-f(x)\|d\mu(x)?$$

• 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? – Pietro Majer Mar 16 at 18:15
• Right good point; I updated the post. – Zorn's Llama Mar 17 at 8:47
• 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 $f_h(x):=f(x-h)$. – Pietro Majer Mar 17 at 15:41
• @Pietro I also figured something like this; but wouldn't we need $f$ to belong to a Sobolev class? – Zorn's Llama Mar 17 at 18:09
• 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 $k\to+\infty$. – Pietro Majer Mar 17 at 18:47