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 $$\dot{W}^{s,p}$$ W ˙ s , p and we call these spaces fractional Paley–Wiener if $$p=2$$ p = 2 and fractional Bernstein spaces if $$p\in (1,\infty )$$ p ∈ ( 1 , ∞ ) , that we denote by $$PW^s_a$$ P W a s and $${\mathcal {B}}^{s,p}_a$$ 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–Pólya inequalities. We conclude by discussing a number of open questions.