Qualitative Korovkin-Type Results on Conservative Approximation

Abstract

In this paper, we present a generalization of the classical Korovkin theorem on positive linear operators. We deduce some convergence results for linear operators defined on C k [0, 1], that preserve some cones of functions related to shape properties. Finally, we show some examples. Academic Press

On saturation effect for linear shape-preserving approximation in Sobolev spaces

“…It is worth noting that non-linear approximation preserving k-monotonicity does not have this shortcoming [10]. On the other hand, for sequences of linear operators preserving k-monotonicity (as well as intersections of cones) there are [5,11] simple convergence conditions (Korovkin type results).…”

Korovkin-type theorem for sequences of operators preserving shape

“…They proved [13] the next Korovkin-type result for a sequences of linear operators preserving shape. . .…”

“…In this paper we extend some results of [13] and prove Korovkin-type results for sequences of linear operators preserving shape-property connected with the cone V h,k (σ ).…”

Linear Relativen-Widths for Linear Operators Preserving an Intersection of Cones

“…Finally, (42) follows from [16], where it is shown (remark after Proposition 1) that there exists a linear operator that maps the cone of positive and concave functions onto the same cone and holds the space 3 .…”