Abstract
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} and we call these spaces fractional Paley–Wiener if p=2 and fractional Bernstein spaces if pin (1,infty ), that we denote by PW^s_a and {mathcal {B}}^{s,p}_a, 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.
Highlights
A renowned theorem due to Paley and Wiener [21] characterizes the entire functions of exponential type a > 0 whose restriction to the real line is square-integrable in terms of the support of the Fourier transform of their restriction to the real line
An analogous characterization holds for entire functions of exponential type a whose restriction to to the real line belongs to some Lp space, p ≠ 2 [9]
Let Ea be the space of entire functions of exponential type a, Ea = f ∈ Hol(C) ∶ for every ε > 0 there exists Cε > 0 such that |f (z)| ≤ Cεe(a+ε)|z|
Summary
A renowned theorem due to Paley and Wiener [21] characterizes the entire functions of exponential type a > 0 whose restriction to the real line is square-integrable in terms of the support of the Fourier transform of their restriction to the real line. In this paper we introduce a family of spaces which generalizes the classical Paley–Wiener and Bernstein spaces; we deal with spaces of entire functions of exponential type a whose restriction to the real line belongs to some homogeneous Sobolev space and we call these spaces fractional Paley–Wiener and Bernstein spaces. We point out that classical results such as sampling theorems for the Paley–Wiener space do not necessarily extend to the fractional setting 8. Remark 1.3 We point out that from the results in the present work we can deduce analogous results for the homogeneous fractional Bernstein spaces Ḃ sa,p , defined as above, but without exponential requiring that Pf0;m;0 = 0 . 4 we investigate the fractional Bernstein spaces proving Theorems 3 and 4, whereas in Sect.
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