Positivity (2017) 21:1189–1199
On two classes of approximation processes of integral
· Ali Aral
· Emre Deniz
Received: 6 June 2016 / Accepted: 25 November 2016 / Published online: 1 December 2016
© Springer International Publishing 2016
Abstract The paper aims to study two classes of linear positive operators represent-
ing modiﬁcations of Picard and Gauss operators. The new operators reproduce both
constants and a given exponential function. Approximation properties in polynomial
weighted spaces are investigated and the speed of convergence is measured using a
certain weighted modulus of smoothness. Also, the asymptotic behavior of the integral
operators are established. Finally, aspects on generalized convexity are analyzed.
Keywords Linear positive operator · Picard operator · Gauss operator · Weighted
space · Voronovskaja formula
Mathematics Subject Classiﬁcation 41A36 · 41A25
The study of the linear methods of approximation, which are given by sequences of
linear positive operators, became a strongly ingrained part of Approximation Theory.
Faculty of Mathematics and Computer Science, Babe¸s-Bolyai University, Street M.
Kog˘alniceanu 1, 400084 Cluj-Napoca, Romania
Department of Mathematics, Faculty of Science and Arts, Kirikkale University, Yahsihan,
71450 Kirikkale, Turkey