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 \(p\in (1,\infty )\), 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.

我们介绍和研究一个包含一个完整函数的空间家族，该变量概括了经典的Paley-Wiener和Bernstein空间。也就是说，我们考虑指数类型为a的所有函数，其对实线的限制属于齐次Sobolev空间\（\ dot {W} ^ {s，p} \），如果\（p = 2 \）和分数伯恩斯坦空间，如果\（p \ in（1，\ infty）\），我们用\（PW ^ s_a \）和\（{\ mathcal {B}} ^ {s，p}表示_一种\）， 分别。对于这些空间，我们提供了Paley–Wiener类型的特征，我们在Hilbert环境中评论了有关采样问题的一些事实，并证明了经典Bernstein和Plancherel–Pólya不等式的推广。最后，我们讨论一些未解决的问题。