Agratini, Octavian An approximation process of Kantorovich type. (English) Zbl 0981.41015 Math. Notes, Miskolc 2, No. 1, 3-10 (2001). A Kantorovich-type modification of the well known operators \[ (R_nf)=\frac{1}{(1+a_n x)^n} \sum_{k=0}^n\binom{n}{k}(a_nx)^kf(\frac{k}{b_n}), \quad x\geq 0,\;n\in \mathbb{N} \] is introduced. In particular, the author changes \(f(\frac{k}{b_n})\) by the integral mean \[ na_n\int_{k/(na_n)}^{(k+1)/(na_n)}f(t) dt. \] The author computes the degree of approximation associated to these operators in certain function spaces. Concretely, he obtains estimations for local \(\text{Lip}_{\alpha}\) functions. The most interesting result of the paper, for this reviewer, is Theorem 4, where some estimations are obtained for functions with jump discontinuities. Reviewer: J.M.Almira (Linares) Cited in 1 ReviewCited in 13 Documents MSC: 41A36 Approximation by positive operators 41A25 Rate of convergence, degree of approximation Keywords:Kantorovich-type operator; Bohman-Korovkin theorem; modulus of smoothness of first order; modulus of variation; local \(Lip_{\alpha}\) functions PDFBibTeX XMLCite \textit{O. Agratini}, Math. Notes, Miskolc 2, No. 1, 3--10 (2001; Zbl 0981.41015) Full Text: EuDML