q-Bernstein polynomials and their iterates |
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Authors: | Sofiya Ostrovska |
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Institution: | Department of Mathematics, Atilim University, 06836 Incek, Ankara, Turkey |
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Abstract: | Let Bn( f,q;x), n=1,2,… be q-Bernstein polynomials of a function f : 0,1]→C. The polynomials Bn( f,1;x) are classical Bernstein polynomials. For q≠1 the properties of q-Bernstein polynomials differ essentially from those in the classical case. This paper deals with approximating properties of q-Bernstein polynomials in the case q>1 with respect to both n and q. Some estimates on the rate of convergence are given. In particular, it is proved that for a function f analytic in {z: |z|<q+} the rate of convergence of {Bn( f,q;x)} to f(x) in the norm of C0,1] has the order q−n (versus 1/n for the classical Bernstein polynomials). Also iterates of q-Bernstein polynomials {Bnjn( f,q;x)}, where both n→∞ and jn→∞, are studied. It is shown that for q(0,1) the asymptotic behavior of such iterates is quite different from the classical case. In particular, the limit does not depend on the rate of jn→∞. |
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Keywords: | q-Bernstein polynomials q-Integers q-Binomial coefficients Convergence Iterates |
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