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

TL;DR

This paper extends Jackson-type approximation estimates to Hilbert spaces and homogeneous manifolds using moduli of continuity and Paley-Wiener spaces, providing a framework for best approximation of vectors.

Contribution

It introduces a new approach to approximation in Hilbert spaces and homogeneous manifolds using semigroup-based moduli of continuity and Paley-Wiener subspaces, establishing Jackson-type inequalities.

Findings

Established Jackson-type estimates for approximation in Hilbert spaces.

Extended approximation results to homogeneous manifolds via Lie group representations.

Defined new moduli of continuity and explored their properties in the context of approximation.

Abstract

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.