Positivity (2011) 15:11–16
Korovkin-type theorem for sequences of operators
S. P. Sidorov
Received: 12 March 2009 / Accepted: 30 November 2009 / Published online: 16 December 2009
© Birkhäuser Verlag Basel/Switzerland 2009
Abstract In the paper we present Korovkin-type theorem concerning conditions of
convergence sequences of linear operators preserving shape.
Keywords Korovkin theorem · Shape-preserving approximation
Mathematics Subject Classiﬁcation (2000) Primary 41A35 · Secondary 41A36
In various applications it is necessary to approximate a function preserving such prop-
erties as monotonicity, convexity, concavity, etc.
The subject of shape-preserving approximation was ﬁrst studied by Shisha  and
by Lorentz and Zeller . This subject was further developed by DeVore and Yu ,
Shvedov [4,5], Newman , Beatson and Leviatan  in their works on monotone
and comonotone approximation. The last 25 years have seen extensive research and
many new results, the most significant of which were summarized in [8–10].
If function f ∈ C[0, 1] possesses certain shape properties, it usually means that
element f belongs to a certain cone V in C[0, 1].
One of the most examined classes of linear operators, which possessed the shape-
preserving property, is linear positive operators. Recall that an operator L deﬁned
This work is supported by RFBR (grant 07-01-00167-a) and the President of the Russian Federation
S. P. Sidorov (
Department of Mechanics and Mathematics, Saratov State University,
Astrakhanskaya 83, 410012 Saratov, Russian Federation