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arXiv · 2005.10074

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

Isaac Z. Pesenson
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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.

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Isaac Z. Pesenson. 2020-05-17. Jackson theorem and modulus of continuity in Hilbert spaces and on homogeneous manifolds. https://doi.org/10.1007/s10476-022-0176-0.pdf

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