Approximation of Functions of Bounded p-variation by Euler Means

In this paper we study the Euler means
e^q_n(f)(x) =∑_k=0ⁿ(nk)q^n−k(1 + q)⁻ⁿS_k(f)(x), q > 0, n ∈ Z+,

where S_k(f) is the k-th partial trigonometric Fourier sum. For p-absolutely continuous functions (f ∈ Cp, 1 < p < ∞) we consider their approximation by the Euler means in uniform and Cp-metric in terms of moduli of continuity ω_k(f)_Cp,k ∈ N, and the best approximations by trigonometric polynomials E_n(f)_Cp. One can note the following inequality for different metrics from Theorem 2
||f - e^q_n(f)||_∞<C₁∑_j=0ⁿ(1+q)^-n(n j)q^n−jE_j(f)Cp, n ∈ N, which is sharp. Also the following generalization of a result due to C. K. Chui and A. S. Holland is proved.

If ω is a modulus of continuity on [0,π], such that δ ∫^π_δ t⁻²ω(t)dt = O(ω(δ)),1 < p < ∞ and f ∈ Cp уsatisfies two properties:1) ω2(f,t)Cp < Cω(t); 2)∫^π_2π/(n+1) t⁻¹||ϕx(t)−ϕx(t+2π/(n+1))||Cp dt = O(ω(1/n)), where ϕx(t) = f(x+t)+f(x−t)−2f(x), то ||e¹_n(f)−f||Cp < Cω(1/n), n ∈ N. Some applications to the approximation in Hölder type metrics are given.