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- W1971507256 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+ε} the rate of convergence of {Bn(f,q;x)} to f(x) in the norm of C[0,1] 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→∞." @default.
- W1971507256 created "2016-06-24" @default.
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- W1971507256 date "2003-08-01" @default.
- W1971507256 modified "2023-09-27" @default.
- W1971507256 title "q-Bernstein polynomials and their iterates" @default.
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- W1971507256 doi "https://doi.org/10.1016/s0021-9045(03)00104-7" @default.
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