Discrete $d$-dimensional moduli of smoothness

Abstract

We show that on the $d$-dimensional cube $I^d\equiv [0,1]^d$ the discrete moduli of smoothness which use only the values of the function on a diadic mesh are sufficient to determine the moduli of smoothness of that function. As an important special case our result implies for $f\in C(I^d)$ and given integer $r$ that when $0<α<r$, the condition \[ \left|Δ^r_{2^{-n} e_i}f\left(\frac{k_1}{2^n},\dots,\frac{k_d}{2^n}\right)\right|\le M2^{-nα} \] for integers $1\le i\le d$, $0\le k_i\le 2^n-r$, $0\le k_j\le 2^n$ when $j\ne i$, and $n=1,2,\dots$ is equivalent to \[ \Bigl|Δ^r_{h u}f(x)\Bigr|\le M_1 h^α\] for $x,u\in\mathbb{R}^d$, $h>0$ and $|u|=1$ such that $x,x+rhu\in I^d$.