Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 
 
 

 

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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. $


References [Enhancements On Off] (What's this?)

  • [Dit84] Z. Ditzian, Moduli of continuity in $ {\bf R}^{n}$ and $ D\subset {\bf R}^{n}$, Trans. Amer. Math. Soc. 282 (1984), no. 2, 611-623. MR 732110 (85c:26007), https://doi.org/10.2307/1999256
  • [Dit87] Z. Ditzian, Moduli of smoothness using discrete data, J. Approx. Theory 49 (1987), no. 2, 115-129. MR 874948 (88c:41036), https://doi.org/10.1016/0021-9045(87)90082-7
  • [Dit88] Z. Ditzian, The modulus of smoothness and discrete data in a square domain, IMA J. Numer. Anal. 8 (1988), no. 3, 311-319. MR 968096 (90c:65017), https://doi.org/10.1093/imanum/8.3.311
  • [RR99] L. J. Ratliff Jr. and D. E. Rush, Triangular powers of integers from determinants of binomial coefficient matrices, Linear Algebra Appl. 291 (1999), no. 1-3, 125-142. MR 1685617 (2000i:11044), https://doi.org/10.1016/S0024-3795(98)10244-6

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 26B35, 41A05, 41A15, 41A25, 41A40, 41A63

Retrieve articles in all journals with MSC (2010): 26B35, 41A05, 41A15, 41A25, 41A40, 41A63


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.

American Mathematical Society