Jackson theorem and modulus of continuity in Hilbert spaces and on homogeneous manifolds
英文摘要：
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.
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