Distribution Frames and Bases

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