论文标题
关于Lebesgue的不平等现象
About Lebesgue inequalities on the classes of generalized Poisson integrals
论文作者
论文摘要
对于功能$ f $，可以用卷积的形式表示$ f（x）= \ frac {a_ {0}} {2}+\ frac {1}π\ int \ limits _ { - π}^π\ sum \ limi ts_ {k = 1}^{\ infty} e^{ - αk^{r}}}} \ cos（kt- \ frac {βπ} {2} {2}）φ（x-t）dt $， $φ\ perp1 $，$α> 0，\ r \ in（0,1）$，$β\ in \ mathbb {r} $，我们建立了form \ begin {equation*}} \ | f-s_ {n-1} {n-1} {n-1}} feben \ begin {equination \ begin {equination*}的lebesgue-type不等式e^{ - αn^{r}} \ left（\ frac {4} {π^{2}}} \ ln \ frac {n^{1-r}} {αr} {αr} +γ_{n} +γ_{n} \ end {equation*}这些不等式均针对所有数字$ n $大于某些数字$ n_ {1} = n_ {1}（α，r）$大的数字$ n $，该$ n_ {1} = n_ {1} $，通过参数$α$和$ r $进行建设性地定义。我们证明存在一个函数，因此可以更改给定估计中的符号“ $ \ leq $”，以“ $ = $”更改。
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 "$=$".
