Approximation of function by Euler means

Abstract

Let 蒖n $$(q, f, x) = \frac{1}{{(1 + q)^n }}\sum\limits_{k = 0}^n {(_k^n )q^{n - k} s_k (f, x)} $$ denote the Euler means of the Fourier series of the 2π-perodic function f(x). For an integer q>0 and a function f(x)∈Hω⊂C([0, 2π]), the main term of deviationf(x)-蒖n(q, f, x) is calculated in this note. Asymptoteaally exact order 3 of decrease of the upper bound of such deviations over the class Hω is also obtained.