Fractional Paley-Wiener and Bernstein spaces

2020/02/27 by Monguzzi, Alessandro, Peloso, Marco M., Salvatori, Maura · 1 citation
#26A33 #30D15 #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2002.12015

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 W^s,p and we call these spaces fractional Paley-Wiener if p=2 and fractional Bernstein spaces if p∈(1,∞), 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.

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• Notes on Bernstein spaces, sampling, Boas interpolation formulas and their extensions to Banach spaces

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