About Lebesgue inequalities on the classes of generalized Poisson integrals

Abstract

For the functions $f$, which can be represented in the form of the convolution $f(x)=\frac{a_0}{2}+\frac{1}{\pi }\underset{-\pi }{\overset{\pi }{\int }}\sum _{k=1}^{\infty }{e}^{-\alpha k^r}\cos (kt-\frac{\beta \pi }{2})\phi (x-t)dt$, $\phi \perp 1$, $\alpha >0,r\in (0,1)$, $\beta \in \mathbb{R}$, we establish the Lebesgue-type inequalities of the form \begin{equation*} \|f-S_{n-1}(f)\|_{C}\leq e^{-αn^{r}}\left(\frac{4}{π^{2}}\ln \frac{n^{1-r}}{αr} + γ_{n} \right) E_{n}(φ)_{C}. \end{equation*} These inequalities take place for all numbers $n$ that are larger than some number $n_1=n_1(\alpha ,r)$, which constructively defined via parameters $\alpha$ and $r$. We prove that there exists a function, such that the sign "$\le$" in given estimate can be changed for "$=$".