Jackson theorem and modulus of continuity in Hilbert spaces and on homogeneous manifolds
Abstract
We consider a Hilbert space H equipped with a set of strongly continuous bounded semigroups satisfying certain conditions. The conditions allow to define a family of moduli of continuity r(s,f),\>r∈ N, s>0, of vectors in H and a family of Paley-Wiener subspaces PWσ parametrized by bandwidth σ>0. These subspaces are explored to introduce notion of the best approximation E(σ, f) of a general vector in H by Paley-Wiener vectors of a certain bandwidth σ>0. The main objective of the paper is to prove the so-called Jackson-type estimate E(σ, f)≤ C( r(σ-1,f)+σ-r\|f\|) for σ>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 H. This way we obtain the Jackson-type estimates on homogeneous manifolds.