Fractional Paley–Wiener and Bernstein spaces

• Monguzzi, Alessandro ^[1] ; Peloso, Marco M. ^[2] ; Salvatori, Maura ^[2]
  1. [1] University of Milano-Bicocca

     University of Milano-Bicocca

     Milán, Italia

     
  2. [2] University of Milan

     University of Milan

     Milán, Italia

     
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 72, Fasc. 3, 2021, págs. 615-643
• Idioma: inglés
• DOI: 10.1007/s13348-020-00303-4
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• Resumen

  • 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 PWsa and Bs,pa, 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.