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Discrete $ d$-dimensional moduli of smoothness


Authors: Z. Ditzian and A. Prymak
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
Published electronically: June 26, 2014
MathSciNet review: 3238430
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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

$\displaystyle \Bigl \vert\Delta ^r_{2^{-n}\boldsymbol {e}_i}f\bigl (\frac {k_1}{2^n},\dots ,\frac {k_d}{2^n}\bigr )\Bigr \vert\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

$\displaystyle \Bigl \vert\Delta ^r_{h\mathbf {e}}f(\mathbf {\xi })\Bigr \vert\l... ...ert=1\textrm { such that}\ \mathbf {\xi },\mathbf {\xi }+rh\mathbf {e}\in I^d. $


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Additional Information

Z. Ditzian
Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, T6G 2G1, Canada
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

DOI: https://doi.org/10.1090/S0002-9939-2014-12088-0
Keywords: Moduli of smoothness, diadic mesh, tensor product splines, Lagrange interpolation.
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
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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