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 a given integer $r$ that when $0<\alpha <r$, the condition \[ \Bigl |\Delta ^r_{2^{-n}\boldsymbol {e}_i}f\bigl (\frac {k_1}{2^n},\dots ,\frac {k_d}{2^n}\bigr )\Bigr |\le M2^{-n\alpha } \] 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 \[ \Bigl |\Delta ^r_{h\mathbf {e}}f(\mathbf {\xi })\Bigr |\le M_1 h^\alpha \quad \textrm {for }\mathbf {\xi },\mathbf {e}\in \mathbb {R}^d,\ h>0 \textrm { and }|\mathbf {e}|=1\textrm { such that}\ \mathbf {\xi },\mathbf {\xi }+rh\mathbf {e}\in I^d. \]