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Zbl 1027.41028
King, J.P.
Positive linear operators which preserve $x^2$. (English)
[J] Acta Math. Hung. 99, No.3, 203-208 (2003). ISSN 0236-5294; ISSN 1588-2632/e

The approximations of continuous functions $f$ on $[0,1]$ by a sequence of positive linear operators ${L_n}$ always converge to $f$ iff $L_n$ preserve the three functions $e_i(x)=x$, $i=0,1,2$ (Korovkin theorem). Replacing the variable $x$ in the Bernstein polynomials by some functions $r_n(x)$ the author defines the operators $L_n$ acting on $\Cal C([0,1])$, satisfying the Korovkin condition and leading to the order of approximation of $f$ at least as good as the order of approximation by Bernstein polynomials. The summability matrix $A$ is defined by means of the functions $r_n(x)$ and it is proved that $A$ preserves the limits of complex sequences provided $\lim_{n\to \infty}r_n(x)=x$.
[Jaczek Gilewicz (Les Arcs sur Argens)]

MSC 2000:: *41A40 Saturation
40G99 Special methods of summability

Keywords: positive linear operators; approximation; summability; Bernstein polynomials

Cited in: Zbl 1183.41025 Zbl 1212.41062 Zbl 1181.41029 Zbl 1164.41337 Zbl 1136.65018 Zbl 1121.41012 Zbl 1121.41013

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