Korovkin-type theorem and application. (English) Zbl 1118.41015

Summary: Let \((L_n)\) be a sequence of positive linear operators on \(C[0,1]\), satisfying that \((L_n(e_i))\) converge in \(C[0,1]\) (not necessarily to \(e_i)\) for \(i=0,1,2\), where \(e_i(x)=x^i\). We prove that the conditions that \((L_n)\) is monotonicity-preserving convexity-preserving and variation diminishing do not suffice to insure the convergence of \((L_n (f))\) for all \(f\in C[0,1]\). We obtain the Korovkin-type theorem and give quantitative results for the approximation properties of the \(q\)-Bernstein operators \(B_{n,q}\) as an application.


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