Problems in the approximation of \(2\pi \)-periodic functions by Fourier sums in the space \(L_2 (2\pi)\) (Q2387805)

Let us denote by \(L_{2} = L_{2} (2\pi )\) the space of square-integrable \(2\pi \)-periodic real-valued functions with the norm \[ \| f \| = \left( {1\over \pi } \int_{\,0}^{\,2\pi } f^{2}(x)\,dx\right)^{1 / 2}. \] For the function \(f \in L_{2} \) we define the Steklov function \[ F_{h} f(x) = f_{h} (x) = {1\over 2h} \int_{x - h}^{x + h} f(t)\,dt, \quad h > 0. \] Further we introduce the differences of order \(k\) as follows: \[ \Delta _{h}^{k} (f;x) = \sum\limits_{i = 0}^{k} ( - 1)^{k - i} {k \choose i} \,F_{h}^{i} f(x), \] where \(F_{h}^{0} f(x) = f(x)\), \(F_{h}^{i} f(x) = F_{h} (F_{h}^{i - 1} f(x)\), \(i = 1\), 2, \dots, \(k\), \(k = 1\), 2, \dots\ The quantity \(\Omega _k (f;\delta ) = \sup\limits_{0<h\leq\delta}\| \Delta_h^k(f;x) \| \) is called \textit{modulus of continuity of order \(k\)} of the function \(f \in L_{2} \). Let \(L_{2}^{r} \) be the class of functions \(f \in L_{2} \) having generalized derivatives \(f^{(i)}(x)\), \(i = 1\), \dots, \(r\), in the sense of Levi, belonging to the space \(L_{2} \), \(L_{2}^{0} = L_{2} \). Further let \(W_{2,\varphi }^{r,k}\) denote the class of functions \(f \in L_{2}^{r} \) for which \(\Omega _{k}(f^{(r)};\delta ) = O[ \varphi (\delta ^k) ]\), \(r \in \mathbb{Z}_+\), \(k \in \mathbb{N}\), where \(\varphi \) is continuous monotone increasing \(2\pi\)-periodic function defined on the interval \([0, + \infty )\) and such that \(\varphi (0) = 0\). At last \(W_{\varphi }^{r,k} \) is the class of continuously differentiable \(2\pi \)-periodic functions for which \[ \sup\limits_{0<h\leq\delta} \{\max \limits_x | \Delta_h^k(f^{(r)};\,x) | \} = O[\varphi (\delta ^{k})], \quad r \in \mathbb{Z}_{ + } ,\quad k \in \mathbb{N}. \] It is evident that \(W_{\varphi }^{r,k} \subset W_{2,\varphi }^{r,k} \). In the present paper the order of the Kolmogorov width of the class \(W_{2,\varphi }^{r,k} \) is calculated. Moreover for the class \(W_{\varphi}^{r,k} \) an estimate of the error of a quadrature formula is obtained.