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We consider a Hilbert space ${\bf H}$ equipped with a set of stronglycontinuous 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 familyof 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-Wienervectors of a certain bandwidth $\sigma>0$. The main objective of the paper isto prove the so-called Jackson-type estimate $\mathcal{E}(\sigma, f)\leqC\left( \Omega^{r}(\sigma^{-1},f)+\sigma^{-r}\|f\|\right)$ for $\sigma>1$.  It was shown in our previous publications that our assumptions are satisfiedfor a strongly continuous unitary representation of a Lie group $G$ in aHilbert space ${\bf H}$. This way we obtain the Jackson-type estimates onhomogeneous manifolds.

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