Abstract
Using the method of Jakimovski and Leviatan from their work in 1969, we construct a general class of linear positive operators. We study the convergence, the evaluation for the rate of convergence in terms of the first modulus of smoothness and we give a Voronovskaja-type theorem for these operators.
Highlights
The aim of this paper is to construct a class of linear operators in more general conditions
The method was inspired by Jakimovski and Leviatan
We do not study the convergence of these operators with the well-known theorem of Bohman-Korovkin
Summary
The aim of this paper is to construct a class of linear operators in more general conditions. We do not study the convergence of these operators with the well-known theorem of Bohman-Korovkin. For a given interval I, we will use the following function sets: B(I) = { f | f : I → R, f bounded on I}, C(I) = { f | f : I → R, f continuous on I}, and CB(I) = B(I) ∩ C(I). For f ∈ CB(I), by the first-order modulus of smoothness of f is meant the function ω( f ; ·) : [0, ∞) → R defined for any δ ≥ 0 by ω( f ; δ) = sup f (x ) − f (x ) : x , x ∈ I, |x − x | ≤ δ
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