Estimates of approximation in terms of a weighted modulus of continuity

Abstract

Consider modulus ω_φ( f, h) = sup { |f(x) − f(y)|| x ≥ 0, y ≥ 0, |x − y| ≤ hφ ( x+y)/2)) }, where φ(x) = x^1/2/1+x^m , x ∈ [0, ∞), m ∈ N, m ≥ 2. We give a characterization of the class of functions f, for which ω_φ( f, h)→∞, (h →∞), and then we obtain quantitative estimates of the approximation of the functions of this class by means of the Sz´asz-Mirakjan operators and by the Baskakov operators.