Sampling formulas for one-parameter groups of operators in Banach spaces

TL;DR

This paper generalizes sampling theories from entire functions to Banach spaces by introducing Bernstein subspaces linked to one-parameter groups of operators, enabling the application of classical sampling results in a new abstract setting.

Contribution

It introduces Bernstein subspaces in Banach spaces associated with one-parameter groups, extending sampling theory to abstract operator settings.

Findings

Defined Bernstein subspaces   for Banach spaces.

Reduced sampling problems for operator trajectories to classical exponential type functions.

Established a framework connecting operator theory with sampling of exponential type functions.

Abstract

We extend some results about sampling of entire functions of exponential type to Banach spaces. By using generator $D$ of one-parameter group $e^{tD}$ of isometries of a Banach space $E$ we introduce Bernstein subspaces $\mathbf{B}_{\sigma}(D),\>\>\sigma>0,$ of vectors $f$ in $E$ for which trajectories $e^{tD}f$ are abstract-valued functions of exponential type which are bounded on the real line. This property allows to reduce sampling problems for $e^{tD}f$ with $f\in \mathbf{B}_{\sigma}(D)$ to known sampling results for regular functions of exponential type $\sigma$.