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\(q\)-Bernstein polynomials and their iterates. (English) Zbl 1093.41013

\(q\)-Bernstein polynomials have been introduced by G. M. Phillips [in Numerical Analysis: A. R. Mitchell 75th Birthday Volume, World Scientific, Singapore, 263–269 (1996)]. For \(q=1\) they reduce to the classical Bernstein polynomials. When \(q\) is in \((0,1)\), the corresponding linear operators are positive; several papers deal with this case. When \(q>1\), the positivity fails. The author shows that in this case the approximation properties of \(q\)-Bernstein polynomials may be better than in the case \(q<1\) or \(q=1\); in particular, for entire functions the rate of convergence is exponential. The iterates of \(q\)-Bernstein operators are also investigated. For \(q>1\), the situation is very similar to the classical case \(q=1\); for \(0<q<1\), it is essentially different.

MSC:

41A36 Approximation by positive operators
33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
41A25 Rate of convergence, degree of approximation
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References:

[1] Andrews, G.E., The theory of partitions, (1976), Addison-Wesley Reading, MA · Zbl 0371.10001
[2] Bernstein, S.N., Démonstration du théorème de Weierstrass fondée sur la calcul des probabilités, Comm. soc. math. charkow Sér. 2, t. 13, 1-2, (1912)
[3] Cooper, S.; Waldron, S., The eigenstructure of the Bernstein operator, J. approx. theory, 105, 133-165, (2000) · Zbl 0963.41006
[4] Goodman, T.N.T.; Oruç, H.; Phillips, G.M., Convexity and generalized Bernstein polynomials, Proc. Edinburgh math. soc., 42, 1, 179-190, (1999) · Zbl 0930.41010
[5] Il’inskii, A.; Ostrovska, S., Convergence of generalized Bernstein polynomials, J. approx. theory, 116, 100-112, (2002) · Zbl 0999.41007
[6] Kelisky, R.P.; Rivlin, T.J., Iterates of Bernstein polynomials, Pacific J. math., 21, 511-520, (1967) · Zbl 0177.31302
[7] Li, X.; Mikusiński, P.; Sherwood, H.; Taylor, M.D., On approximation of copulas, () · Zbl 0905.60015
[8] Lorentz, G.G., Bernstein polynomials, (1986), Chelsea New York
[9] A. Lupaş, A q-analogue of the Bernstein operator, University of Cluj-Napoca, Seminar on Numerical and Statistical Calculus, preprint No. 9, 1987.
[10] Micchelli, C., The saturation class and iterates of the Bernstein polynomials, J. approx. theory, 8, 1-18, (1973) · Zbl 0258.41012
[11] Oruç, H.; Phillips, G.M., A generalization of Bernstein polynomials, Proc. Edinburgh math. soc., 42, 2, 403-413, (1999) · Zbl 0930.41009
[12] Oruç, H.; Tuncer, N., On the convergence and iterates of q-Bernstein polynomials, J. approx. theory, 117, 301-313, (2002) · Zbl 1015.33012
[13] Petrone, S., Random Bernstein polynomials, Scand. J. statist., 26, 3, 373-393, (1999) · Zbl 0939.62046
[14] Phillips, G.M., On generalized Bernstein polynomials, (), 263-269
[15] Phillips, G.M., Bernstein polynomials based on the q-integers, Ann. numer. math., 4, 511-518, (1997) · Zbl 0881.41008
[16] Phillips, G.M., A de casteljau algorithm for generalized Bernstein polynomials, Bit, 37, 1, 232-236, (1997) · Zbl 0877.65007
[17] Phillips, G.M., A generalization of the Bernstein polynomials based on the q-integers, Anziam j., 42, 79-86, (2000) · Zbl 0963.41005
[18] Titchmarsh, E.C., Theory of functions, (1986), Oxford University Press Oxford · Zbl 0601.10026
[19] Videnskii, V.S., Bernstein polynomials, (1990), Leningrad State Pedagocical University Leningrad, (Russian)
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