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

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 $\sigma$.