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Distribution Frames and Bases

  • Camillo Trapani
  • Salvatore Triolo
  • Francesco TschinkeEmail author
Article

Abstract

In this paper we will consider, in the abstract setting of rigged Hilbert spaces, distribution valued functions and we will investigate conditions for them to constitute a ”continuous basis” for the smallest space \(\mathcal D\) of a rigged Hilbert space. This analysis requires suitable extensions of familiar notions as those of frames, Riesz bases and orthonormal bases. A motivation for this study comes from the Gel’fand–Maurin theorem which states, under certain conditions, the existence of a family of generalized eigenvectors of an essentially self-adjoint operator on a domain \(\mathcal D\) which acts like an orthonormal basis of the Hilbert space \(\mathcal H\). The corresponding object will be called here a Gel’fand distribution basis. The main results are obtained in terms of properties of a conveniently defined synthesis operator.

Keywords

Distributions Rigged Hilbert spaces Frames Bases 

Mathematics Subject Classification

Primary 47A70 Secondary 42C15 42C30 

Notes

Acknowledgements

The authors thank the referees for their useful comments and suggestions. This paper has been made within the framework of the Project INdAM-GNAMPA 2018 Alcuni aspetti di teoria spettrale di operatori e di algebre; frames in spazi di Hilbert rigged.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • Camillo Trapani
    • 1
  • Salvatore Triolo
    • 2
  • Francesco Tschinke
    • 1
    Email author
  1. 1.Dipartimento di Matematica e InformaticaUniversità di PalermoPalermoItaly
  2. 2.Dipartimento DEIMUniversità di PalermoPalermoItaly

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