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 , , and , the inequality , , is satisfied, where is a piecewise polynomial of degree 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.