The aim of this paper is to overview the notions of distribution frames and bases, the main results on them and to indicate perspectives of further studies, highlighting what differentiates them from the usual notions in Hilbert space. Far from being a purely mathematical generalization, some examples provide connection with Gel’fand theory of generalized eigenvectors of self-adjoint or normal operators and open the way to some applications in the mathematical formulation of quantum mechanics.
Trapani, C., Tschinke, F. (2026). Frames and Bases Beyond Hilbert Space: Overview and Perspectives. COMPLEX ANALYSIS AND OPERATOR THEORY, 20 [10.1007/s11785-026-02055-8].
Frames and Bases Beyond Hilbert Space: Overview and Perspectives
Camillo Trapani
;Francesco Tschinke
2026-10-10
Abstract
The aim of this paper is to overview the notions of distribution frames and bases, the main results on them and to indicate perspectives of further studies, highlighting what differentiates them from the usual notions in Hilbert space. Far from being a purely mathematical generalization, some examples provide connection with Gel’fand theory of generalized eigenvectors of self-adjoint or normal operators and open the way to some applications in the mathematical formulation of quantum mechanics.
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