论文标题
杰克逊定理和希尔伯特空间中连续性的模量
Jackson theorem and modulus of continuity in Hilbert spaces and on homogeneous manifolds
论文作者
论文摘要
我们考虑一个Hilbert Space $ {\ bf H} $，配备了一组强烈连续的有界半群，可满足某些条件。 条件允许定义一个连续性$ω^{r}（s，f），\> r \ in \ mathbb {n}，s> 0，$ {\ bf h} $中的向量和vectors $的家族，以及一个paley-wiener subspaces $pw_σ$ parameTriped bandwiend $ parameTried bandwiend $ 0 $ 0 $ 0。 探索了这些子空间，以引入最佳近似值$ \ MATHCAL {本文的主要目的是证明所谓的杰克逊型估计$ \ mathcal {e}（σ，f）\ leq c \ left（ω^{r}（ω^{r}（σ^{ - 1}，f），f）+σ^{ - r} { - r} { - r} \ | f \ | f \ | f \ | \ | f \ | \ | f \ | f \ | \ | for $ for $ for $ for $σ> 1 $。 在我们之前的出版物中显示，我们的假设对于希尔伯特太空中的谎言组$ g $的强烈连续统一表示满足。这样，我们就可以获得杰克逊型统一的估计。
We consider a Hilbert space ${\bf H}$ equipped with a set of strongly continuous bounded semigroups satisfying certain conditions. The conditions allow to define a family of moduli of continuity $Ω^{r}(s,f),\>r\in \mathbb{N}, s>0,$ of vectors in ${\bf H}$ and a family of Paley-Wiener subspaces $PW_σ$ parametrized by bandwidth $σ>0$. These subspaces are explored to introduce notion of the best approximation $\mathcal{E}(σ, f)$ of a general vector in ${\bf H}$ by Paley-Wiener vectors of a certain bandwidth $σ>0$. The main objective of the paper is to prove the so-called Jackson-type estimate $\mathcal{E}(σ, f)\leq C\left( Ω^{r}(σ^{-1},f)+σ^{-r}\|f\|\right)$ for $σ>1$. It was shown in our previous publications that our assumptions are satisfied for a strongly continuous unitary representation of a Lie group $G$ in a Hilbert space ${\bf H}$. This way we obtain the Jackson-type estimates on homogeneous manifolds.
