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arXiv · 1812.08472

Distributions Frames and bases

Camillo Trapani·Salvatore Triolo·Francesco Tschinke
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Abstract

In this paper we will consider, in the abstract setting of rigged Hilbert spaces, distribution valued functions and we will investigate, in particular, 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 frame, Riesz basis and orthonormal basis. 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 {\em Gel'fand distribution basis}. The main results are obtained in terms of properties of a conveniently defined {\em synthesis operator}.

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Camillo Trapani, Salvatore Triolo, Francesco Tschinke. 2018-12-20. Distributions Frames and bases. https://doi.org/10.1007/s00041-018-09659-5

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