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arXiv · 2005.13849

About Lebesgue inequalities on the classes of generalized Poisson integrals

A. S. Serdyuk·T. A. Stepaniuk
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Abstract

For the functions $f$, which can be represented in the form of the convolution $f(x)=\frac{a_{0}}{2}+\frac{1}π\int\limits_{-π}^π\sum\limits_{k=1}^{\infty}e^{-αk^{r}}\cos(kt-\frac{βπ}{2})φ(x-t)dt$, $φ\perp1$, $α>0, \ r\in(0,1)$, $β\in\mathbb{R}$, 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 $n_{1}=n_{1}(α,r)$, which constructively defined via parameters $α$ and $r$. We prove that there exists a function, such that the sign "$\leq$" in given estimate can be changed for "$=$".

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A. S. Serdyuk, T. A. Stepaniuk. 2020-05-28. About Lebesgue inequalities on the classes of generalized Poisson integrals. https://arxiv.org/abs/2005.13849

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