Abstract
The aim of this paper is to present new non-asymptotic norm estimates in C[0,1] for the q-Bernstein operators B n,q in the case q > 1. While for 0 < q ≤ 1, ∥B n,q ∥ = 1 for all n ∈ ℕ, in the case q > 1, the norm ∥B n,q ∥ grows rather rapidly as n → +∞ and q → +∞. Both theoretical and numerical comparisons of the new estimates with the previously available ones are carried out. The conditions are determined under which the new estimates are better than the known ones.
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References
G.E. Andrews, R. Askey, R. Roy, Special Functions, Cambridge Univ. Press., Cambridge, 1999.
L. C. Biedenharn, The quantum group SU q (2) and a q-analogue of the boson operators, J.Phys.A: Math. Gen. 22, 873–878 (1989).
Ch. A. Charalambides, The q-Bernstein basis as a q-binomial distribution, J. Stat. Planning and Inference 140 (8), 2184–2190 (2010).
A.II’inskii, S.Ostrovska, Convergence of generalized Bernstein polynomials, J. Approx. Theory, 116 (1), 100–112 (2002).
S. Jing, The q-deformed binomial distribution and its asymptotic behaviour, J. Phys. A: Math. Gen. 27, 493–499 (1994).
N. Mahmudov, The moments for q-Bernstein operators in the case 0 < q < 1, Numer. Algorithms, 53 (4), 439–450 (2010).
I. Ya. Novikov, Asymptotics of the roots of Bernstein polynomials used in the construction of modified Daubechies wavelets, Mathematical Notes, 71 (1–2), 217–229 (2002).
S. Ostrovska, q-Bernstein polynomials and their iterates, J. Approx. Th., 123 (2), 232–255 (2003).
S. Ostrovska, The first decade of the q-Bernstein polynomials: results and perspectives, J. Math. Anal. Appr. Th., 2 (1), 35–51 (2007).
S. Ostrovska, A. Y. Özban, The norm estimates of the q-Bernstein operators for varying q > 1, Computers & Mathematics with Applications 62 (12), 4758–4771 (2011).
M. I. Ostrovskii, Regularizability of inverse linear operators in Banach spaces with bases, Siberian Math. J., 33 (3), 470–476 (1992).
G.M. Phillips, Interpolation and Approximation by Polynomials, Springer-Verlag, 2003.
M. M. Popov, Complemented subspaces and some problems of the modern geometry of Banach spaces (Ukrainian), Mathematics today, Fakt, Kiev, 13, 78–116 (2007).
V.S.Videnskii, On some classes of q-parametric positive operators, Operator Theory: Advances and Applications, 158, 213–222 (2005).
H. Wang, S. Ostrovska, The norm estimates for the q-Bernstein operator in the case q > 1, Math. Comp., 79, 353–363 (2010).
Z. Wu, The saturation of convergence on the interval [0,1] for the q-Bernstein polynomials in the case q > 1, J. Math. Anal. Appl., 357 (1), 137–141 (2009).
M. Zeiner, Convergence properties of the q-deformed binomial distribution, Applicable Analysis and Discrete Mathematics, 4 (1), 66–80 (2010).
Acknowledgement
We would like to express our sincere gratitude to Mr. P. Danesh from Atilim University, Academic Writing and Advisory Centre, for his help in the preparation of the manuscript.
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Ostrovska, S., Özban, A.Y. (2013). Non-asymptotic Norm Estimates for the q-Bernstein Operators. In: Anastassiou, G., Duman, O. (eds) Advances in Applied Mathematics and Approximation Theory. Springer Proceedings in Mathematics & Statistics, vol 41. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6393-1_24
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DOI: https://doi.org/10.1007/978-1-4614-6393-1_24
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