  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.
