Language:   Search:   Contact
Zentralblatt MATH has released its new interface!
For an improved author identification, see the new author database of ZBMATH.

Query:
Fill in the form and click »Search«...
Format:
Display: entries per page entries
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

Highlights
Master Server