On Some Classes of q-parametric Positive linear Operators
- Victor S. Videnskii
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Recently, G.M. Phillips ,  has introduced the q-parametric Bernstein polynomials denoted by B n (f, x, q), 0 < q ≤ 1. In  the investigation of the properties of these polynomials is based on some generalization of divided differences and on the Newton interpolation formula. A. Il’inskii and S. Ostrovska  have considered linear operators B ∞ (f, x, q) which were derived by means of B n (f,x, q) with the help of an informal passing to the limit for n → ∞. In this paper we give another more natural proof of the main results of , . Moreover, for the Voronovskaya’s asymptotic formula we obtain the estimate of the remainder term. We also consider the modifi- cation of these operators in order to improve the degree of approximation of twice differentiable functions.
- S.N. Bernstein, Demonstration du théoreme de Weierstrass fondée sur la calcul des probabilités, Comm. Soc. Math. Charkow Ser. 2, Vol. 13 (1912), 1–2. Collected works. Vol. 1, 1952, 106–107.
- S.N. Bernstein, Complément à l’article de E. Voronovskaya, Détermination de la forme asymptotique de l’approximation de function par les polynômes de M. Bernstein, Dokl. AN USSR, Ser. A, No. 4 (1932), 86–92. Collected works. Vol. 2, 1954, 155–158.
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- V.S. Videnskii, Linear positive operators of finite rank, Leningrad, 1985. (Russian)
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- On Some Classes of q-parametric Positive linear Operators
- Book Title
- Selected Topics in Complex Analysis
- Book Subtitle
- The S. Ya. Khavinson Memorial Volume
- pp 213-222
- Print ISBN
- Online ISBN
- Series Title
- Operator Theory: Advances and Applications
- Series Volume
- Birkhäuser Basel
- Copyright Holder
- Birkhäuser Verlag
- Additional Links
- q-Bernstein polynomials
- central moments
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- Editor Affiliations
- 30. Department of Higher Mathematics, Moscow State Civil Engineering University
- Victor S. Videnskii (31)
- Author Affiliations
- 31. Dept. of Mathematics, Russian State Pedagogical University, Moika 48, Saint-Petersburg, 191186, Russia
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