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

Abstract

We consider a Hilbert space $\mathbf{H}$ 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 $\mathbf{H}$ and a family of Paley-Wiener subspaces $P{W}_{\sigma }$ 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 $\mathbf{H}$ 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 $\mathcal{E}(\sigma ,f)\le C\left({\Omega }^r({\sigma }^{-1},f)+{\sigma }^{-r}\Vert f\Vert \right)$ 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 $\mathbf{H}$. This way we obtain the Jackson-type estimates on homogeneous manifolds.

arXiv commentarXiv admin note: text overlap with arXiv:1708.07416