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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Nearly monotone spline approximation in $ \mathbb{L}_p$


Authors: K. Kopotun, D. Leviatan and A. V. Prymak
Journal: Proc. Amer. Math. Soc. 134 (2006), 2037-2047
MSC (2000): Primary 41A10, 41A25, 41A29
Published electronically: December 19, 2005
MathSciNet review: 2215773
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Abstract: It is shown that the rate of $ \mathbb{L}_p$-approximation of a non-decreasing function in $ \mathbb{L}_p$, $ 0<p<\infty$, by ``nearly non-decreasing" splines can be estimated in terms of the third classical modulus of smoothness (for uniformly spaced knots) and third Ditzian-Totik modulus (for Chebyshev knots), and that estimates in terms of higher moduli are impossible. It is known that these estimates are no longer true for ``purely" monotone spline approximation, and properties of intervals where the monotonicity restriction can be relaxed in order to achieve better approximation rate are investigated.


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Additional Information

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

D. Leviatan
Affiliation: School of Mathematical Sciences, Raymond and Beverley Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, 69978, Israel
Email: leviatan@post.tau.ac.il

A. V. Prymak
Affiliation: Faculty of Mechanics and Mathematics, National Taras Shevchenko University of Kyiv, Kyiv, 01033, Ukraine
Email: prymak@univ.kiev.ua

DOI: http://dx.doi.org/10.1090/S0002-9939-05-08365-6
PII: S 0002-9939(05)08365-6
Keywords: Monotone approximation by piecewise polynomials and splines, degree of approximation, Jackson type estimates
Received by editor(s): February 11, 2005
Published electronically: December 19, 2005
Additional Notes: The first author was supported in part by NSERC of Canada.
Part of this work was done while the third author visited Tel Aviv University in May 2004
Communicated by: Jonathan M. Borwein
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.