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 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.
Fractional Paley–Wiener and Bernstein spaces
