Nearly monotone spline approximation in

Authors:
K. Kopotun, D. Leviatan and A. V. Prymak

Journal:
Proc. Amer. Math. Soc. **134** (2006), 2037-2047

MSC (2000):
Primary 41A10, 41A25, 41A29

DOI:
https://doi.org/10.1090/S0002-9939-05-08365-6

Published electronically:
December 19, 2005

MathSciNet review:
2215773

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: It is shown that the rate of -approximation of a non-decreasing function in , , 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.

**1.**R. A. DeVore, Y. K. Hu, and D. Leviatan,*Convex polynomial and spline approximation in ,*, Constr. Approx.**12**(1996), 409-422. MR**1405006 (97j:41008)****2.**R. A. DeVore, D. Leviatan, and X. M. Yu,*Polynomial approximation in*, Constr. Approx.**8**(1992), 187-201. MR**1152876 (93f:41011)****3.**R. A. DeVore and G. G. Lorentz, Constructive Approximation, Springer-Verlag, Berlin, 1993. MR**1261635 (95f:41001)****4.**Y. K. Hu, K. A. Kopotun, and X. M. Yu,*Weak copositive and intertwining approximation*, J. Approx. Theory**96**(1999), 213-236. MR**1671196 (2000a:41026)****5.**D. Leviatan and I. A. Shevchuk,*Nearly comonotone approximation*, J. Approx. Theory**95**(1998), 53-81. MR**1645976 (99j:41012)****6.**D. Leviatan and I. A. Shevchuk,*Some positive results and counterexamples in comonotone approximation II*, J. Approx. Theory**100**(1999), 113-143. MR**1710556 (2000f:41026)****7.**D. J. Newman,*The Zygmund condition for polygonal approximation*, Proc. Amer. Math. Soc.**45**(1974), 303-304. MR**0361553 (50:13998)****8.**D. J. Newman, E. Passow, and L. Raymon,*Piecewise monotone polynomial approximation*, Trans. Amer. Math. Soc.**172**(1972), 465-472. MR**0310506 (46:9604)****9.**J. A. Roulier,*Nearly comonotone approximation*, Proc. Amer. Math. Soc.**47**(1975), 84-88. MR**0364967 (51:1220)****10.**A. S. Shvedov,*Orders of coapproximation of functions by algebraic polynomials*, Mat. Zametki**29**(1981), 117-130; Eng. transl. Math. Notes**30**(1981), 63-70. MR**0604156 (82c:41009)**

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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:
https://doi.org/10.1090/S0002-9939-05-08365-6

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.