Sampling Formulas for Groups of Operators in Banach Spaces

摘要

We extend some results about sampling of entire functions of exponen­tial type to Banach spaces. By using the generator D of a one-parameter group e~(tD) of isometries of a Banach space E we introduce Bernstein sub-spaces B_σ(D), σ ＞ 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∈ Bσ(D)) to known sampling results for regular functions of exponential type σ.