## Univariate splines: Equivalence of moduli of smoothness and applications

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- by Kirill A. Kopotun;
- Math. Comp.
**76**(2007), 931-945 - DOI: https://doi.org/10.1090/S0025-5718-06-01920-X
- Published electronically: November 27, 2006
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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.## References

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## Bibliographic Information

**Kirill A. Kopotun**- Affiliation: Department of Mathematics, University of Manitoba, Winnipeg, Manitoba, R3T 2N2, Canada
- Email: kopotunk@cc.umanitoba.ca
- 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.
- © Copyright 2006
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2291843