Fractional Paley–Wiener and Bernstein spaces

AbstractWe introduce and study a family of spaces of entire functions in one variable that generalise the classical Paley–Wiener and Bernstein spaces. Namely, we consider entire functions of exponential typeawhose restriction to the real line belongs to the homogeneous Sobolev space$$\dot{W}^{s,p}$$W˙s,pand we call these spaces fractional Paley–Wiener if$$p=2$$p=2and fractional Bernstein spaces if$$p\in (1,\infty )$$p∈(1,∞), that we denote by$$PW^s_a$$PWasand$${\mathcal {B}}^{s,p}_a$$Bas,p, respectively. For these spaces we provide a Paley–Wiener type characterization, we remark some facts about the sampling problem in the Hilbert setting and prove generalizations of the classical Bernstein and Plancherel–Pólya inequalities. We conclude by discussing a number of open questions.