×

Coconvex approximation. (English) Zbl 1057.41012

Summary: Let \(f\in\mathbb{C}[-1,1]\) change its convexity finitely many times, in the interval. We are interested in estimating the degree of approximation of \(f\) by polynomials which are coconvex with it, namely, polynomials that change their convexity exactly at the points where \(f\) does. We discuss some Jackson-type estimates where the constants involved depend on the location of the points of change of convexity. We also show that in some cases the constants may be taken independent of the points of change of convexity, but that in other cases this dependence is essential. But mostly we obtain such estimates for functions \(f\) that themselves are continuous piecewise polynomials on the Chebyshev partition, which form a single polynomial in a small neighborhood of each point of change of convexity. These estimates involve the \(k\) modulus of smoothness of the piecewise polynomials when they themselves are of degree \(k-1\).

MSC:

41A29 Approximation with constraints
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] DeVore, R. A., Monotone approximation by polynomials, SIAM J. Math., 8, 906-921 (1977) · Zbl 0368.41002
[2] DeVore, R. A.; Yu, X. M., Pointwise estimates for monotone polynomial approximation, Constr. Approx., 1, 323-331 (1985) · Zbl 0583.41006
[3] Ditzian, Z.; Totik, V., Moduli of Smoothness (1987), Springer-Verlag: Springer-Verlag New York · Zbl 0666.41001
[4] Dzyadyk, V. K., Introduction to the Theory of Uniform Approximation of Functions by Polynomials (1977), Nauka: Nauka Moscow · Zbl 0481.41001
[5] Dzyubenko, G. A.; Gilewicz, J.; Shevchuk, I. A., Piecewise monotone pointwise approximation, Constr. Approx., 14, 311-348 (1998) · Zbl 0912.41007
[6] Gilewicz, J.; Shevchuk, I. A., Comonotone approximation, Fund. Prikl. Mat., 2, 319-363 (1996) · Zbl 0908.41003
[7] Kopotun, K.; Leviatan, D.; Shevchuk, I. A., The degree of coconvex polynomial approximation, Proc. Amer. Math. Soc., 127, 409-415 (1999) · Zbl 0914.41005
[8] Leviatan, D., Shape-preserving approximation by polynomials, J. Comput. Appl. Math., 121, 73-94 (2000) · Zbl 0969.41007
[9] Leviatan, D.; Shevchuk, I. A., Constants in comonotone polynomial approximation—A survey, (Müller, M. W.; Buhmann, M. D.; Mache, D. H.; Felten, M., New Developments in Approximation Theory (1999), Birkhäuser: Birkhäuser Basel) · Zbl 0938.41002
[10] Leviatan, D.; Shevchuk, I. A., Some positive results and counterexamples in comonotone approximation II, J. Approx. Theory, 100, 113-143 (1999) · Zbl 0934.41013
[11] Pleshakov, M. G.; Shatalina, A. V., Piecewise coapproximation and Whitney inequality, J. Approx. Theory, 105, 189-210 (2000) · Zbl 1101.41307
[12] Shevchuk, I. A., Polynomial Approximation and Traces of Functions Continuous on a Segment (1992), Naukova Dumka: Naukova Dumka Kyiv · Zbl 0782.41013
[13] Shvedov, A. S., Orders of coapproximation of functions by algebraic polynomials, Mat. Zametki, 29, 117-130 (1981) · Zbl 0462.41006
[14] Wu, X.; Zhou, S. P., A counterexample in comonotone approximation in Lp space, Colloq. Math., 64, 265-274 (1993) · Zbl 0894.41009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.