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

Trapani C., Triolo S., Tschinke F. (2019). Distribution Frames and Bases. JOURNAL OF FOURIER ANALYSIS AND APPLICATIONS, 25(4), 2109-2140 [10.1007/s00041-018-09659-5].

Distribution Frames and Bases

Trapani C.;Triolo S.;Tschinke F.
2019-01-01

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 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
Settore MAT/05 - Analisi Matematica
Trapani C., Triolo S., Tschinke F. (2019). Distribution Frames and Bases. JOURNAL OF FOURIER ANALYSIS AND APPLICATIONS, 25(4), 2109-2140 [10.1007/s00041-018-09659-5].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10447/371264
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