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Definitions and Notation:

Fix a positive constant $M>0$ with positive integers $m,n$ and the standard orthonormal basis $e_1,\dots,e_n$ of $\mathbb{R}^n$.

For every positive integer $N$, define the class $\mathcal{H}^{(m,n,N)}$ of piecewise constant functions of the form

$$ \sum_{j=1}^N\, k_j\cdot \prod_{i=1}^n I_{[a_{i,j},b_{i,j}]}\big(\langle x,e_i\rangle\big), $$ where $k_1,\dots,k_N\in \mathbb{R}^m$.

Question:

  1. At what rate can $\mathcal{H}^{(m,n,N)}$ approximate continuous functions? I am looking for a rate $r>0$ such that for every sufficiently large $N$, and for every uniformly continuous $f:[-M,M]^n\rightarrow \mathbb{R}^m$ with modulus of continuity $\omega$, $$ \inf_{h\in \mathcal{H}^{(m,n,N)}}\, \sup_{x\in [-M,M]^n}\, \|h(x)-f(x)\|_2 < cN^{-r} $$ where $c$ may depend on $M,m,n,\omega$ but is independent of $f$ and $N$.
  2. Can we achieve a better rate if $f \in C^k(\mathbb{R}^n,\mathbb{R}^m)$ and we know it's modulus of smoothness?
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    $\begingroup$ I think the key example is $m=1$, $f= w \sum \langle x,e_i \rangle$, with a minimum error achieved by dividing the big cube into smaller cubes all of the same size. If so, the limiting error is $2MN^{-1/n}nw$ and the desired rate is $r=1/n$. $\endgroup$
    – user44143
    Commented Oct 4, 2022 at 15:03
  • $\begingroup$ More generally, for a uniformly continuous $f$ with mod. of c. $\omega$, the distance from $f$ to $H^{(m,n,N)}$ should be around $\omega(MN^{-1/n}n^{1/2})$ (taking the locally constant function equal to $f$ in the center of each of $N$ cubes of equal size, and pretending $N$ is a power of $n$). So I’d say to get a distance $cN^{-r}$ from $H^{(m,n,N)}$ you need $f$ to be Hoelder of exponent $rn$. So $r=1/n$ should be ok, but for $f$ Lipschitz. $\endgroup$
    – Pietro Majer
    Commented Oct 4, 2022 at 21:29
  • $\begingroup$ Matt and Pietro; Do you know of a reference? $\endgroup$
    – ABIM
    Commented Oct 5, 2022 at 1:56
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    $\begingroup$ I don’t know a reference — but why define piecewise constancy with box-shaped pieces? $\endgroup$
    – user44143
    Commented Oct 5, 2022 at 9:26
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    $\begingroup$ It seems the quantity you define on the LHS, (that is the point-set distance from $f$ to the space $H^{m,n,N}$) is itself a sort of "mean modulus of continuity", something weaker than the usual (uniform) modulus of continuity: it could actually be $cN^{-r}$ even if $f$ is not Hoelder, e.g. if it has a very thin, non Hoelder cuspid at a single point $x_0$ (then the cubes of the approximating h have to be very small around $x_0$, a lot of them, but that is possible if elsewhere f is more flat so that less cubes are needed) $\endgroup$
    – Pietro Majer
    Commented Oct 5, 2022 at 21:15

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