Jackson theorem and modulus of continuity in Hilbert spaces and on homogeneous manifolds

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 $\Omega^{r}(s,f),\>r\in \mathbb{N}, s>0,$ of vectors in ${\bf H}$ and a family of Paley-Wiener subspaces $PW_{\sigma}$ parametrized by bandwidth $\sigma>0$. These subspaces are explored to introduce notion of the best approximation $\mathcal{E}(\sigma, f)$ of a general vector in ${\bf H}$ by Paley-Wiener vectors of a certain bandwidth $\sigma>0$. The main objective of the paper is to prove the so-called Jackson-type estimate $\mathcal{E}(\sigma, f)\leq C\left( \Omega^{r}(\sigma^{-1},f)+\sigma^{-r}\|f\|\right)$ for $\sigma>1$. 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.