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Pointwise estimates for higher order convexity preserving polynomial approximation

Published online by Cambridge University Press:  17 February 2009

Jia-Ding Cao  and
Heinz H. Gonska
Jia-Ding Cao
Affiliation:
Dept of Mathematics, Fudan University, Shanghai, PRC.
Heinz H. Gonska
Affiliation:
Dept of Mathematics, University of Duisburg, D-47048 Duisburg, Germany.
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Abstract

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DeVore-Gopengauz-type operators have attracted some interest over the recent years. Here we investigate their relationship to shape preservation. We construct certain positive convolution-type operators Hn, s, j which leave the cones of j-convex functions invariant and give Timan-type inequalities for these. We also consider Boolean sum modifications of the operators Hn, s, j show that they basically have the same shape preservation behavior while interpolating at the endpoints of [−1, 1], and also satisfy Telyakovskiῐ- and DeVore-Gopengauz-type inequalities involving the first and second order moduli of continuity, respectively. Our results thus generalize related results by Lorentz and Zeller, Shvedov, Beatson, DeVore, Yu and Leviatan.

Type
Research Article
Information
The ANZIAM Journal , Volume 36 , Issue 2 , October 1994 , pp. 213 - 233
Copyright
Copyright © Australian Mathematical Society 1994

References

[1]Beatson, R. K., “Joint approximation of a function and its derivatives”, in Approximation theory III (Proc. Int. Sympos. Austin 1980) (ed. Cheney, E. W.), (Acad. Press, New York, 1980) 199–206.Google Scholar
[2]Berens, H. and DeVore, R. A., “A characterization of Bernstein polynomials”, in Approximation theory III (Proc. Int. Sympos. Austin 1980) (ed. Cheney, E. W.), (Acad. Press, New York, 1980) 213–219.Google Scholar
[3]Berens, H. and Lorentz, G. G., “Inverse theorems for Bernstein polynomials”, Indiana Univ. Math. J. 21 (1972) 693–708.CrossRefGoogle Scholar
[4]Cao, Jia-ding, “On linear approximation methods (Chinese)”, Acta Sci. Natur. Univ. Fudan 9 (1964) 43–52.Google Scholar
[5]Cao, Jia-ding, “Generalization of Timan's theorem, Lehnhoff's theorem and Telyakovskiῐ's theorem”, Volume SM-DU-106 of Schriftenreihe des Fachbereichs Mathematik, (Universität Duisburg, 1986).Google Scholar
[6]Cao, Jia-ding, “Generalizations of Timan theorem, Lehnhoff theorem and Telyakovskiῐ theorem (Chinese)”, Kexue Tongbao 15 (1986) 1132–1135, Science Bulletin (English Edition) 32 (1987), 1225–1229.Google Scholar
[7]Cao, Jia-ding and Gonska, H. H., “Approximation by Boolean sums of positive linear operators”, Rend. Mat. 6 (1986) 525–546.Google Scholar
[8]Cao, Jia-ding and Gonska, H. H., “Approximation by Boolean sums of positive linear operators II: Gopengauz-type estimates”, J. Approx. Theory 57 (1989) 77–89.CrossRefGoogle Scholar
[9]Cao, Jia-ding and Gonska, H. H., “Pointwise estimates for modified positive linear operators”, Portugal. Math. 46 (1989) 401–430.Google Scholar
[10]Cao, Jia-ding and Gonska, H. H., “Approximation by Boolean sums of positive linear operators III: Estimates for some numerical approximation schemes”, Numer. Funct. Anal. Optim. 10 (1989) 643–672.CrossRefGoogle Scholar
[11]Cao, Jia-ding and Gonska, H. H., “Computation of DeVore-Gopengauz-type approximants”, in Approximation Theory VI (Proc. Int. Sympos. College Station 1989) (eds. Chui, C. K. et al. ), (Acad. Press, New York, 1989) 117–120.Google Scholar
[12]Cao, Jia-ding and Gonska, H. H., “Approximation by Boolean sums of linear operators: Telyakovskiῐ-type estimates”, Bull. Austral. Math. Soc. 42 (1990) 253–266.CrossRefGoogle Scholar
[13]Cao, Jia-ding and Gonska, H. H., “Approximation by Boolean sums of positive linear operators IV: shape preservation by DeVore-Gopengauz-type approximants”, Volume SIAI-EBS-2–92 of Schriftenreihe des Instituts für Angewandte Informatik, (European Business School, 1992).Google Scholar
[14]Chalmers, B. L. and Taylor, G. D., “Uniform approximation with constraints”, Jahresber. Deutsch. Math.-Verein. 81 (1979) 49–86.Google Scholar
[15]DeVore, R. A., The approximation of continuous functions by positive linear operators (Springer, Berlin-Heidelberg-New York, 1972).CrossRefGoogle Scholar
[16]DeVore, R. A. and Yu, Xiang-ming, “Pointwise estimates for monotone polynomial approximation”, Constr. Approx. 1 (4) (1985) 323–331.CrossRefGoogle Scholar
[17]Farin, G., Curves and surfaces for computer aided geometric design, 2nd ed. (Acad. Press, New York, 1990).Google Scholar
[18]Gonska, H. H., “On Pičugov-Lehnhoff operators”, Volume SM-DU-86 of Schriftenreihe des Fachbereichs Mathematik, (Universität Duisburg, 1985).Google Scholar
[19]Gonska, H. H., “Modified Pičugov-Lehnhoff operators”, in Approximation theory V (Proc. Int. Sympos., College Station 1986) (eds. Chui, C. K. et al. ), (Acad. Press, New York, 1986) 355–358.Google Scholar
[20]Hoschek, J. and Lasser, D., Grundlagen der geometrischen Datenverarbeitung, 2nd ed. (Teubner, Stuttgart, 1992).CrossRefGoogle Scholar
[21]Isaacson, E. and Keller, H. B., Analysis of numerical methods (John Wiley and Sons, New York, 1966).Google Scholar
[22]Leviatan, D., “Pointwise estimates for convex polynomial approximation”, Proc. Amer. Math. Soc. 98 (1986) 471–474.CrossRefGoogle Scholar
[23]Lorentz, G. G., “Inequalities and the saturation classes of Bernstein polynomials”, in On Approximation Theory (Proc. Conf. Oberwolfach 1963) (eds. Butzer, P. L. et al. ), (Birkhäuser, Basel, 1964) 200–207.Google Scholar
[24]Lorentz, G. G., Jetter, K. and Riemenschneider, S. D., Birkhoff interpolation (Addison-Wesley, Reading/MA, 1983).Google Scholar
[25]Lorentz, G. G. and Zeller, K. L., “Degree of approximation by monotone polynomials I”, J. Approx. Theory 1 (1968) 501–504.CrossRefGoogle Scholar
[26]Matsuoka, Y., “On the approximation of functions by some singular integrals”, Tôhoku Math. J. 18 (1966) 13–43.CrossRefGoogle Scholar
[27]Pᾰltᾰnea, R., “On the estimate of the pointwise approximation of functions by linear positive functionals”, Volume 35 of Studia Univ. Babeş-Bolyai Math., (1990) 11–24.Google Scholar
[28]Roulier, J., “Linear operators invariant on the nonnegative monotone functions”, SIAM J. Numer. Anal. 8 (1971) 30–35.CrossRefGoogle Scholar
[29]Senderovizh, R., “On convexity preserving operators”, S1AM J. Math. Anal. 9 (1978) 157–159.Google Scholar
[30]Shvedov, A. S., “Jackson's theorem in Lp 0 < p < 1, for algebraic polynomials and orders of comonotone approximations (Russian)”, Mat. Zametki 25 (1) (1979) 107–117, Engl, transl.: Math. Notes 25 (1979), 57–63.Google Scholar
[31]Shvedov, A. S., “Orders of coapproximation of functions by algebraic polynomials (Russian)”, Mat. Zametki 29 (1) (1981) 117–130, Engl, transl.: Math. Notes 29 (1981), 63–70.Google Scholar
[32]Yu, Xiang-ming, “Pointwise estimates for convex polynomial approximation”, Approx. Theory Appl. 1 (4) (1985) 65–74.Google Scholar
[33]Zygmund, A., Trigonometric series (Cambridge Univ. Press, Cambridge, 1959).Google Scholar