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
Full Text: DOI


[1] Devore, R. A.; Lorentz, G. G., Constructive Approximation (1993), Springer: Springer Berlin · Zbl 0797.41016
[2] Goodman, T. N.T.; Phillips, G. M., Convexity and generalized Bernstein polynomials, Proc. Edinburgh Math. Soc., 42, 1, 179-190 (1999) · Zbl 0930.41010
[3] II’inskii, A.; Ostrovska, S., Convergence of generalized Bernstein polynomials, J. Approx. Theory, 116, 1, 100-112 (2002) · Zbl 0999.41007
[4] Oruc, H.; Phillips, G. M., A generalization of the Bernstein polynomials, Proc. Edinburgh Math. Soc., 42, 2, 403-413 (1999) · Zbl 0930.41009
[5] Oruc, H.; Tuncer, N., On the convergence and iterates of \(q\)-Bernstein polynomials, J. Approx. Theory, 117, 2, 301-313 (2002) · Zbl 1015.33012
[6] Ostrovska, S., \(q\)-Bernstein polynomials and their iterates, J. Approx. Theory, 123, 2, 232-255 (2003) · Zbl 1093.41013
[7] Phillips, G. M., Bernstein polynomials based on the \(q\)-integers, Ann. Numer. Math., 4, 511-518 (1997) · Zbl 0881.41008
[8] Tiberiu, T., Meyer-König and Zeller operators based on the \(q\)-integers, Rev. Anal. Numer. Theory Approx., 29, 2, 221-229 (2000) · Zbl 1023.41022
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.