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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?

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?

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: 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 $N$.

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?

Definitions and Notation:

Fix positive integers $n,m$ and a positive constant $M>0$.
Consider the standard basis orthonormal $e_1,\dots,e_n$ of $\mathbb{R}^n$.
For every positive integer $N$, define the class $\mathcal{H}^{(N,n,m)}$ 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 $k_1,\dots,k_N\in \mathbb{R}^m$ and $-M\le a_{i,n}<b_{i,n}\le M$ for every $i=1,\dots,n$ and $j=1,\dots,N$.

Question: Let $f:\mathbb{R}^n\rightarrow \mathbb{R}^m$ be a uniformly continuous function and restricted to a cube $[-M,M]^n$ with modulus of continuity $\omega$. Are these known "best approximation" rates of the class $\mathcal{H}^{(N,n,m)}$ know? That is, is there a known quantitative relationship between $f$'s modulus of continuity $\omega$, $M,n$, and $m$ and a rate $r>0$ such that: for every sufficiently large $N\in \mathbb{N}_+$ have that $$ \inf_{h\in \mathcal{H}^{(N,m,n)}}\, \sup_{x\in [-M,M]^n}\, \|h(x)-f(x)\|_2 \lesssim N^{-r}. $$

Note: We hide a non-negative constant independent of $N$ by $\lesssim$.

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: 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 $N$.