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Distributions Frames and bases. (arXiv:1812.08472v1 [math.FA])
来源于:arXiv
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}. 查看全文>>