Corpus ID: 218972042

About Lebesgue inequalities on the classes of generalized Poisson integrals

@article{Serdyuk2020AboutLI,
  title={About Lebesgue inequalities on the classes of generalized Poisson integrals},
  author={A. Serdyuk and T. Stepaniuk},
  journal={arXiv: Classical Analysis and ODEs},
  year={2020}
}
For the functions $f$, which can be represented in the form of the convolution $f(x)=\frac{a_{0}}{2}+\frac{1}{\pi}\int\limits_{-\pi}^{\pi}\sum\limits_{k=1}^{\infty}e^{-\alpha k^{r}}\cos(kt-\frac{\beta\pi}{2})\varphi(x-t)dt$, $\varphi\perp1$, $\alpha>0, \ r\in(0,1)$, $\beta\in\mathbb{R}$, we establish the Lebesgue-type inequalities of the form \begin{equation*} \|f-S_{n-1}(f)\|_{C}\leq e^{-\alpha n^{r}}\left(\frac{4}{\pi^{2}}\ln \frac{n^{1-r}}{\alpha r} + \gamma_{n} \right) E_{n}(\varphi)_{C… Expand

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