Jackson theorem and modulus of continuity for unitary representations of Lie groups

For a strongly continuous unitary representation of a Lie group $G$ in a Hilbert space ${\bf H}$ we consider an analog of the Laplace operator $L$ and use it to define subspaces of Paley-Wiener vectors $PW_{\sigma}(L)$. It allows 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 group representation is used to introduce a family of moduli of continuity $\Omega^{r}(s,f),\>r\in \mathbb{N}, s>0,$ of vectors in ${\bf H}$. The main objective of the paper is to prove the so-called Jackson-type estimate $\mathcal{E}(\sigma, f)\leq C \Omega^{r}(\sigma^{-1},f)$.