Exact Constant of Approximation of Periodic Functions by Cesaro Means

Abstract

In the paper, we present new findings concerning Cesaro-type operators. Special attention is given to the Cesaro \((C, \alpha)\) summation operators, which form a widely used family of summation methods in Fourier analysis. We establish sharp inequality for the upper bound of uniform deviations of Cesaro \((C, \alpha)\) summation operators of the second order for the class of continuous periodic functions.   Let $n\in \mathbb N$ and $\displaystyle  x_k^{(n)} = x_{k-1}^{(n)} + {2\pi}/{(2n+1)},$\ $k\in\{ 0, \pm1, \ldots, \pm(n-1), n\},$ be the points  from the interval $[-\pi, \pi]$ such that  $\displaystyle  -\pi \leq x_{-n}^{(n)} < x_{-n+1}^{(n)} < \ldots < x_{-1}^{(n)} < x_0^{(n)} < x_1^{(n)} < \ldots < x_n^{(n)} \leq \pi;$  the set of points $\{x_k^{(n)}\}$ is uniquely determined by the value of  $x_0^{(n)}$. The Ces\`{a}ro $(C, \alpha)$ summation operators are defined by   $\displaystyle  \sigma_n^{(\alpha)}[f]\left( \left\{x_k^{(n)}\right\}; x\right) = \frac{2}{2 n+1} \sum_{k=-n}^n f\left(x_k^{(n)}\right) K_n^{\alpha}\left(x - x_k^{(n)}\right),$ $K_n^{(\alpha)}(t)=\frac{1}{A_n^\alpha}\sum_{\nu=0}^n A_{n-\nu}^{\alpha-1} D_\nu (t),$   where  $  D_\nu(t)=\frac{\sin\left(\left(\nu+{1}/{2}\right)t\right)}{2 \sin\left(t/2\right)}$ is the Dirichlet kernel, and   $ A_n^\alpha = \frac{(\alpha + 1) \ldots (\alpha + n)}{n!}$ $(n \in \mathbb N),$\ $A_0^\alpha = 1,$\ are the Cesaro numbers,  $\alpha > -1$. Let $\mathbb{T} = [-\pi, \pi]$ and $\mathrm C(\mathbb T)$ be the space of continuouson $\mathbb{T}$ functions with the norm $\|f\|_{\mathrm C} = \max\{|f(t)|\colon t\in \mathbb T\}.$   The main result is contained in the following statement (Theorem 1): Let $f\in \mathrm C(\mathbb T)$. Then the inequality  $$ \left\|f - \sigma^{(2)}_n[f]\left(\left\{x_k^{(n)}\right\}\right)\right\|_{\mathrm C} \leq \frac{11}{9\ln2}\cdot \ln(n+1) \cdot\omega\left(f; {2\pi}/{(2n+1)}\right),\ n\in \mathbb N, $$ holds. The constant ${11}/{(9\ln2)}$  in this inequality is sharp.