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On the sets of convergence for sequences of the \(q\)-Bernstein polynomials with \(q > 1\). (English) Zbl 1254.41013
Summary: The aim of this paper is to present new results related to the convergence of the sequence of the \(q\)-Bernstein polynomials \(\{B_{n, q}(f; x)\}\) in the case \(q > 1\), where \(f\) is a continuous function on \([0 ,1]\). It is shown that the polynomials converge to \(f\) uniformly on the time scale \(\mathbb J_q = \{q^{-j}\}^\infty_{j=0} \cup \{0\}\), and that this result is sharp in the sense that the sequence \(\{B_{n, q}(f; x)\}^\infty_{n=1}\) may be divergent for all \(x \in R \setminus \mathbb J_q\). Further, the impossibility of the uniform approximation for the Weierstrass-type functions is established. Throughout the paper, the results are illustrated by numerical examples.

MSC:
41A25 Rate of convergence, degree of approximation
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[1] G. E. Andrews, R. Askey, and R. Roy, Special Functions, vol. 71, Cambridge University Press, Cambridge, UK, 1999. · Zbl 1147.76535
[2] S. Ostrovska, “The first decade of the q-Bernstein polynomials: results and perspectives,” Journal of Mathematical Analysis and Approximation Theory, vol. 2, no. 1, pp. 35-51, 2007. · Zbl 1159.41301
[3] N. Mahmudov, “The moments of q-Bernstein operators in the case 0<q<1,” Numerical Algorithms, vol. 53, no. 4, pp. 439-450, 2010. · Zbl 1198.41009
[4] M. Popov, “Narrow operators (a survey),” in Function Spaces IX, vol. 92, pp. 299-326, Banach Center Publications, 2011. · Zbl 1244.47020
[5] P. Sabancıgil, “Higher order generalization of q-Bernstein operators,” Journal of Computational Analysis and Applications, vol. 12, no. 4, pp. 821-827, 2010. · Zbl 1197.41020
[6] H. Wang, “Properties of convergence for \omega ,q-Bernstein polynomials,” Journal of Mathematical Analysis and Applications, vol. 340, no. 2, pp. 1096-1108, 2008. · Zbl 1144.41004
[7] Z. Wu, “The saturation of convergence on the interval [0,1] for the q-Bernstein polynomials in the case q>1,” Journal of Mathematical Analysis and Applications, vol. 357, no. 1, pp. 137-141, 2009. · Zbl 1236.41011
[8] X.-M. Zeng, D. Lin, and L. Li, “A note on approximation properties of q-Durrmeyer operators,” Applied Mathematics and Computation, vol. 216, no. 3, pp. 819-821, 2010. · Zbl 1188.41015
[9] C. A. Charalambides, “The q-Bernstein basis as a q-binomial distribution,” Journal of Statistical Planning and Inference, vol. 140, no. 8, pp. 2184-2190, 2010. · Zbl 1191.60014
[10] S. C. Jing, “The q-deformed binomial distribution and its asymptotic behaviour,” Journal of Physics A, vol. 27, no. 2, pp. 493-499, 1994. · Zbl 0820.60011
[11] I. Ya. Novikov and S. B. Stechkin, “Fundamentals of wavelet theory,” Russian Mathematical Surveys, vol. 53, no. 6, pp. 53-128, 1998. · Zbl 0955.42019
[12] M. I. Ostrovskiĭ, “Regularizability of inverse linear operators in Banach spaces with a basis,” Siberian Mathematical Journal, vol. 33, no. 3, pp. 470-476, 1992.
[13] V. S. Videnskii, “On some classes of q-parametric positive linear operators,” in Selected Topics in Complex Analysis, vol. 158, pp. 213-222, Birkhäuser, Basel, Switzerland, 2005. · Zbl 1088.41008
[14] A. Il’inskii and S. Ostrovska, “Convergence of generalized Bernstein polynomials,” Journal of Approximation Theory, vol. 116, no. 1, pp. 100-112, 2002. · Zbl 0999.41007
[15] S. Ostrovska, “q-Bernstein polynomials and their iterates,” Journal of Approximation Theory, vol. 123, no. 2, pp. 232-255, 2003. · Zbl 1093.41013
[16] S. Ostrovska and A. Y. Özban, “The norm estimates of the q-Bernstein operators for varying q>1,” Computers & Mathematics with Applications, vol. 62, no. 12, pp. 4758-4771, 2011. · Zbl 1236.41026
[17] S. Ostrovska, “On the approximation of analytic functions by the q-Bernstein polynomials in the case q>1,” Electronic Transactions on Numerical Analysis, vol. 37, pp. 105-112, 2010. · Zbl 1207.41004
[18] I. J. Schoenberg, “On polynomial interpolation at the points of a geometric progression,” Proceedings of the Royal Society of Edinburgh, vol. 90, no. 3-4, pp. 195-207, 1981. · Zbl 0506.41003
[19] A. Pinkus, “Weierstrass and approximation theory,” Journal of Approximation Theory, vol. 107, no. 1, pp. 1-66, 2000. · Zbl 0968.41001
[20] G. G. Lorentz, Bernstein Polynomials, Chelsea, New York, NY, USA, 2nd edition, 1986. · Zbl 0989.41504
[21] S. Cooper and S. Waldron, “The eigenstructure of the Bernstein operator,” Journal of Approximation Theory, vol. 105, no. 1, pp. 133-165, 2000. · Zbl 0963.41006
[22] R. A. DeVore and G. G. Lorentz, Constructive Aapproximation, vol. 303, Springer, Berlin, Germany, 1993. · Zbl 0797.41016
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