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Univariate splines: Equivalence of moduli of smoothness and applications


Author: Kirill A. Kopotun
Journal: Math. Comp. 76 (2007), 931-945
MSC (2000): Primary 65D07, 41A15, 26A15; Secondary 41A10, 41A25, 41A29
DOI: https://doi.org/10.1090/S0025-5718-06-01920-X
Published electronically: November 27, 2006
MathSciNet review: 2291843
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Abstract: Several results on equivalence of moduli of smoothness of univariate splines are obtained. For example, it is shown that, for any $ 1\leq k\leq r+1$, $ 0\leq m\leq r-1$, and $ 1\leq p\leq\infty$, the inequality $ n^{-\nu} \omega_{k-\nu }(s^{(\nu)}, n^{-1})_p \sim \omega_{k} (s, n^{-1})_p$, $ 1\leq \nu \leq \min\{ k, m+1\}$, is satisfied, where $ s\in \mathbb{C}^m[-1,1]$ is a piecewise polynomial of degree $ \leq r$ on a quasi-uniform (i.e., the ratio of lengths of the largest and the smallest intervals is bounded by a constant) partition of an interval. Similar results for Chebyshev partitions and weighted Ditzian-Totik moduli of smoothness are also obtained. These results yield simple new constructions and allow considerable simplification of various known proofs in the area of constrained approximation by polynomials and splines.


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Additional Information

Kirill A. Kopotun
Affiliation: Department of Mathematics, University of Manitoba, Winnipeg, Manitoba, R3T 2N2, Canada
Email: kopotunk@cc.umanitoba.ca

DOI: https://doi.org/10.1090/S0025-5718-06-01920-X
Keywords: Univariate splines, moduli of smoothness, degree of approximation, Jackson type estimates, polynomial and spline approximation
Received by editor(s): June 1, 2005
Received by editor(s) in revised form: August 25, 2005
Published electronically: November 27, 2006
Additional Notes: The author was supported in part by NSERC of Canada.
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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