6533b823fe1ef96bd127e313
RESEARCH PRODUCT
Distributions Frames and bases
Francesco TschinkeCamillo TrapaniSalvatore Triolosubject
Pure mathematicsGeneral Mathematics02 engineering and technologyBaseDistributionSpace (mathematics)01 natural sciencessymbols.namesakeSettore MAT/05 - Analisi MatematicaGeneralized eigenvector0202 electrical engineering electronic engineering information engineeringFOS: MathematicsFrameOrthonormal basisRigged Hilbert spaces0101 mathematicsMathematicsBasis (linear algebra)Applied MathematicsOperator (physics)010102 general mathematics47A70 42C15 42C30Hilbert space020206 networking & telecommunicationsRigged Hilbert spaceFunctional Analysis (math.FA)Mathematics - Functional AnalysisDistribution (mathematics)symbolsAnalysisdescription
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}.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2018-12-20 |