Discrete $d$-dimensional moduli of smoothness
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- by Z. Ditzian and A. Prymak PDF
- Proc. Amer. Math. Soc. 142 (2014), 3553-3559 Request permission
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 \[ \Bigl |\Delta ^r_{h\mathbf {e}}f(\mathbf {\xi })\Bigr |\le M_1 h^\alpha \quad \textrm {for }\mathbf {\xi },\mathbf {e}\in \mathbb {R}^d,\ h>0 \textrm { and }|\mathbf {e}|=1\textrm { such that}\ \mathbf {\xi },\mathbf {\xi }+rh\mathbf {e}\in I^d. \]References
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Additional Information
- Z. Ditzian
- Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, T6G 2G1, Canada
- MR Author ID: 58415
- Email: zditzian@math.ualberta.ca, zditzian@shaw.ca
- A. Prymak
- Affiliation: Department of Mathematics, University of Manitoba, Winnipeg, Manitoba, R3T 2N2, Canada
- Email: prymak@gmail.com
- Received by editor(s): April 20, 2012
- Received by editor(s) in revised form: November 5, 2012
- Published electronically: June 26, 2014
- Additional Notes: The second author was supported by NSERC of Canada.
- Communicated by: Walter Van Assche
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 142 (2014), 3553-3559
- MSC (2010): Primary 26B35; Secondary 41A05, 41A15, 41A25, 41A40, 41A63
- DOI: https://doi.org/10.1090/S0002-9939-2014-12088-0
- MathSciNet review: 3238430