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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.

MSC:
 41A36 Approximation by positive operators 47B65 Positive linear operators and order-bounded operators
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References:
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