DOI: 10.2478/s12175-013-0162-x Math. Slovaca 63 (2013), No. 5, 1153–1161 GLOBAL SMOOTHNESS PRESERVATION WITH SECOND ORDER MODULUS OF SMOOTHNESS Gancho Tachev (Communicated by J´ an Bors´ ık ) ABSTRACT. We establish the global smoothness preservation of a function f by the sequence of linear positive operators. Our estimate is in terms of the second order Ditzian-Totik modulus of smoothness. Application is given to the Bernstein operator. c 2013 Mathematical Institute Slovak Academy of Sciences 1. Introduction Over the recent years there has been considerable interest in the preservation of global smoothness properties by linear operators. This concept was intro- duced in [4], a recent book on the subject is [5]. Statements concerning global smoothness preservaton can be of various types. The first statements on this topic have been established as ω s (B n f,ε) ≤ M · ω s (f,ε), ε ≥ 0, where ω s is the classical sth order modulus of smoothness. That is, for f ∈ C[0, 1]: ω 1 (f,ε) := sup  |f (x) - f (y)| : |x - y|≤ ε  ω s (f,ε) := sup 0<h≤ε max 0≤x≤1-sh |∆ s h f (x)|, where ∆ s h f (x) is the sth symmetric difference of the function f . For every function f ∈ C[0, 1] the Bernstein polynomial operator is given by B n (f ; x)= n  k=0 f  k n  ·  n k  x k (1 - x) n-k , x ∈ [0, 1]. 2010 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n: Primary 41A10, 41A15, 41A17, 41A36. K e y w o r d s: Bernstein polynomials, global smoothness preservation, Ditzian-Totik moduli of smoothness. Unauthenticated Download Date | 7/29/18 2:27 PM