On the degree of approximation of functions in C2π1 with operators of the Jackson type

AbstractLet C2π1 be the class of real functions of a real variable that are 2π-periodic and have a continuous derivative. The positive linear operators of the Jackson type are denoted by Ln,p(n ∈ N), where p is a fixed positive integer. The object of this paper is to determine the exact degree of approximation when approximating functions f ϵ C2π1 with the operators Ln,p. The value of maxx¦Ln,p(f x) − f(x)¦ is estimated in terms of ω1(f; δ), the modulus of continuity of f′, with δ = πn. Exact constants of approximation are obtained for the operators Ln,p (n ∈ N, p ≥ 2) and for the Fejér operators Ln,1 (n ∈ N). Furthermore, the limiting behaviour of these constants is investigated as n → ∞, and p → ∞, separately or simultaneously