Positivity (2016) 20:25–40
The new forms of Voronovskaya’s theorem in weighted
· Ali Aral
· Ioan Rasa
Received: 9 October 2014 / Accepted: 23 April 2015 / Published online: 8 May 2015
© Springer Basel 2015
Abstract The Voronovskaya theorem which is one of the most important pointwise
convergence results in the theory of approximation by linear positive operators (l.p.o)
is considered in quantitative form. Most of the results presented in this paper mainly
depend on the Taylor’s formula for the functions belonging to weighted spaces. We
ﬁrst obtain an estimate for the remainder of Taylor’s formula and by this estimate we
give the Voronovskaya theorem in quantitative form for a class of sequences of l.p.o.
The Grüss type approximation theorem and the Grüss-Voronovskaya-type theorem in
quantitative form are obtained as well. We also give the Voronovskaya type results for
the difference of l.p.o acting on weighted spaces. All results are also given for well-
known operators, Szasz-Mirakyan and Baskakov operators as illustrative examples.
Our results being Voronovskaya-type either describe the rate of pointwise convergence
or present the error of approximation simultaneously.
Keywords Voronovskaya theorem · Grüss-type-Voronovskaya theorem ·
Weighted modulus of continuity · Difference of operators
Mathematics Subject Classiﬁcation 41A25 · 41A36
Department of Mathematics, Faculty of Science and Arts, Kirikkale University, 71450 Yahsihan,
Technical University of Cluj-Napoca, Cluj-Napoca, Romania