Abstract
Since in the case q > 1, q-Bernstein polynomials are not positive linear operators on C[0,1], the study of their approximation properties is essentially more difficult than that for 0<q<1. Despite the intensive research conducted in the area lately, the problem of describing the class of functions in C[0,1] uniformly approximated by their q-Bernstein polynomials (q > 1) remains open. It is known that the approximation occurs for functions admit ting an analytic continuation into a disc {z:|z| < R}, R > 1. For functions without such an assumption, no general results on approximation are available. In this paper, it is shown that the function f(x) = ln (x + a), a > 0, is uniformly approximated by its q-Bernstein polynomials (q > 1) on the interval [0,1] if and only if a ≥ 1.
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References
Bernstein, S.N.: Démonstration du théorème de Weierstrass fondée sur la calcul des probabilités. Comm. Soc. Math. Charkow t.13(série 2), 1–2 (1912)
Cooper, S., Waldron, S.: The eigenstructure of the Bernstein operator. J. Approx. Theory 105, 133–165 (2000)
Derriennic, M.-M.: Modified Bernstein polynomials and Jacobi polynomials in q-calculus. Rend. Circ. Mat. Palermo Suppl. 76(Serie II), 269–290 (2005)
DeVore, R., Lorentz, G.G.: Constructive Approximation. Springer, Berlin Heidelberg New York (1993)
Goodman, T.N.T., Oruç, H., Phillips, G.M.: Convexity and generalized Bernstein polynomials. Proc. Edinb. Math. Soc. 42(1), 179–190 (1999)
Il’inskii, A.: A probabilistic approach to q-polynomial coefficients, Euler and Stirling numbers I. Mat. Fiz. Anal. Geom. 11(4), 434–448 (2004)
Il’inskii, A., Ostrovska, S.: Convergence of generalized Bernstein polynomials. J. Approx. Theory 116, 100–112 (2002)
Lorentz, G.G.: Bernstein Polynomials. Chelsea, New York (1986)
Lupaş, A.: A q-analogue of the Bernstein operator. University of Cluj-Napoca, Seminar on numerical and statistical calculus, No. 9 (1987)
Nevanlinna, R.: Analytic Functions. Springer, Berlin Heidelberg New York (1970)
Oruç, H., Tuncer, N.: On the convergence and iterates of q-Bernstein polynomials. J. Approx. Theory 117, 301–313 (2002)
Ostrovska, S.: q-Bernstein polynomials and their iterates. J. Approx. Theory 123, 232–255 (2003)
Ostrovska, S.: On the q-Bernstein polynomials. Adv. Stud. Contemp. Math. 11(2), 193–204 (2005)
Ostrovska, S.: On the improvement of analytic properties under the limit q-Bernstein operator. J. Approx. Theory 138, 37–53 (2006)
Ostrovska, S.: The approximation by q-Bernstein polynomials in the case q↓1. Arch. Math. 86(3), 282–288 (2006)
Phillips, G.M.: On generalized Bernstein polynomials. In: Griffits, D.F., Watson, G.A. (eds.) Numerical Analysis: A.R. Mitchell 75th Birthday, volume, pp. 263–269. World Science, Singapore (1996)
Phillips, G.M.: Bernstein polynomials based on the q-integers. Ann. Numer. Math. 4, 511–518 (1997)
Phillips, G.M.: Interpolation and Approximation by Polynomials. Springer, Berlin Heidelberg New York (2003)
Titchmarsh, E.C.: Theory of Functions. Oxford University Press, London (1986)
Videnskii, V.S.: On some classes of q-parametric positive operators. Oper. Theory Adv. Appl. 158, 213–222 (2005)
Videnskii, V.S.: On the polynomials with respect to the generalized Bernstein basis. In: Problems of Modern Mathematics and Mathematical Education, Hertzen Readings, pp. 130–134. St.-Petersburg, Russia (2005) (Russian)
Wang, H.: Korovkin-type theorem and application. J. Approx. Theory 132(2), 258–264 (2005)
Wang, H., Meng, F.: The rate of convergence of q-Bernstein polynomials for 0 < q < 1. J. Approx. Theory 136(2), 151–158 (2005)
Wang, H.: Voronovskaya type formulas and saturation of convergence for q-Bernstein polynomials for 0 < q < 1. J. Approx. Theory 145(2), 182–195 (2007)
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Ostrovska, S. The approximation of logarithmic function by q-Bernstein polynomials in the case q > 1. Numer Algor 44, 69–82 (2007). https://doi.org/10.1007/s11075-007-9081-7
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DOI: https://doi.org/10.1007/s11075-007-9081-7