Fractional Paley-Wiener and Bernstein spaces

We 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 type a whose restriction to the real line belongs to the homogeneous Sobolev space W-s,W-p and we call these spaces fractional Paley-Wiener if p = 2 and fractional Bernstein spaces if p is an element of (1, infinity), that we denote by PWas and B-a(s,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-Polya inequalities. We conclude by discussing a number of open questions.