Approximation of continuous functions by Euler means (Q1085366)

Let \({\mathcal E}_ n(1,f,x)=\frac{1}{2^ n}\sum^{n}_{k=1}C^ k_ nS_ k(x,f)\) denote the Euler means of the Fourier series of the \(2\pi\)- periodic function f(x). For a function \((fx)\in H^{\omega (\delta)}\subset C([0,2\pi])\) the main term of deviation f(x)-\({\mathcal E}_ n(1,f,x)\) is calculated. The asymptotically exact order of decrease of the upper bound of such deviations over the class \(H^{\omega (\delta)}\) is also obtained.