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.
Categories: math.FA