Discrete $d$-dimensional moduli of smoothness

@article{Ditzian2014DiscreteM,
  title={Discrete \$d\$-dimensional moduli of smoothness},
  author={Z. Ditzian and A. Prymak},
  journal={arXiv: Numerical Analysis},
  year={2014}
}
  • Z. Ditzian, A. Prymak
  • Published 2014
  • Mathematics
  • arXiv: Numerical Analysis
  • 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 $0 0$ and $|u|=1$ such that $x,x+rhu\in I^d$. 

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