Transformation of the second order modulus by positive linear operators
Abstract We obtain estimates for the transformation of the second order modulus of continuity by positive linear operators which satisfy certain conditions, which are common to a large class of approximation operators.
On a Problem of Gonska
This paper investigates global smoothness preservation by the Bernstein operators. When the smoothness is measured by the modulus of continuity, this problem is well understood. When it is measured by the second order modulus of smoothness, H. Gonska conjectured that the Lipschitz classes of second order keep invariate under the Bernstein operators. Here we present a counterexample to this conjecture. Then we introduce a new modulus of smoothness and show that the Lip-α(0 < α ≤ 1) classes measured by this modulus are invariate under the Bernstein operators. By means of this modulus we also improve some previous results concerning global smoothness preservation.
A NONSMOOTH EXTENSION FOR THE BERNSTEIN-STANCU OPERATORS AND AN APPLICATION
D.D. Stancu defined in (8) a class of approximation opera- tors which are more general than the well-known Bernstein operators. We define here a new type of approximation operators which extend the Bernstein-Stancu operators. These new operators have the advantage that the points where the given function f : (0,1) ! R is calculated can be independently chosen in each interval of the equidistant division of the interval (0,1). Moreover, all possible such choices of intermediary points cover the whole interval (0,1). Finally, we consider a particular case as an application.