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Zbl 1179.41024
Mahmudov, Nazim I.
Korovkin-type theorems and applications. (English)
[J] Cent. Eur. J. Math. 7, No. 2, 348-356 (2009). ISSN 1895-1074; ISSN 1644-3616/e

Summary: Let $\{T_n\}$ be a sequence of linear operators on $C[0,1]$, satisfying that $\{T_n (e_i)\}$ converge in $C[0,1]$ (not necessarily to $e_i$ ) for $i = 0,1,2$, where $e_i = t^i$ . We prove Korovkin-type theorem and give quantitative results on $C^{2}[0,1]$ and $C[0,1]$ for such sequences. Furthermore, we define King's type $q$-Bernstein operator and give quantitative results for the approximation properties of such operators.

MSC 2000:: *41A36 Approximation by positive operators
47B65 Positive and order bounded operators

Keywords: Korovkin approximation; positive operator; $q$-Bernstein operators; King's type $q$-Bernstein operator; $q$-operators

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Zentralblatt MATH Berlin [Germany]