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.
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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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Communicated by Hans G. Feichtinger.
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Trapani, C., Triolo, S. & Tschinke, F. Distribution Frames and Bases. J Fourier Anal Appl 25, 2109–2140 (2019). https://doi.org/10.1007/s00041-018-09659-5
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DOI: https://doi.org/10.1007/s00041-018-09659-5