Journal
JOURNAL OF FOURIER ANALYSIS AND APPLICATIONS
Volume 25, Issue 4, Pages 2109-2140Publisher
SPRINGER BIRKHAUSER
DOI: 10.1007/s00041-018-09659-5
Keywords
Distributions; Rigged Hilbert spaces; Frames; Bases
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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 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 D which acts like an orthonormal basis of the Hilbert space 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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