## 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.

### MSC:

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

### Keywords:

Korovkin-type theorem; $$q$$-Bernstein operators
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### References:

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