# Distributions Frames and bases

## TL;DR

This paper explores the concept of distribution-valued functions as continuous bases within rigged Hilbert spaces, extending classical notions like frames and bases, motivated by the Gel'fand-Maurin theorem and introducing Gel'fand distribution bases.

## Contribution

It introduces the concept of Gel'fand distribution bases and extends classical basis notions to the setting of rigged Hilbert spaces, providing new theoretical insights.

## Key findings

- Characterization of distribution-valued functions as continuous bases.
- Extension of frame, Riesz basis, and orthonormal basis notions.
- Main results via properties of a synthesis operator.

## Abstract

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}.