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Korovkin-type theorems and applications. (English) Zbl 1179.41024
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.

41A36 Approximation by positive operators
47B65 Positive linear operators and order-bounded operators
Full Text: DOI
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