Riesz-Fischer maps, Semiframes and Frames in rigged Hilbert spaces

TL;DR

This paper reviews and extends the theory of frames, bases, and Riesz-Fischer maps in distribution spaces, providing new results and examples within the context of rigged Hilbert spaces and tempered distributions.

Contribution

It introduces new results on Riesz-Fischer maps and semi-frames in distribution spaces, expanding the understanding of frames in rigged Hilbert spaces.

Findings

New results on Riesz-Fischer maps and semi-frames.

Examples in tempered distribution spaces.

A comprehensive survey of frames and bases in distribution spaces.

Abstract

In this note a review, some considerations and new results about maps with values in a distribution space and domain in a $\sigma$-finite measure space $X$, are obtained. In particular, it is a survey about Bessel, frames and bases (in particular Riesz and Gel'fand bases) in a distribution space. In this setting, the Riesz-Fischer maps and semi-frames are defined and new results about them are attained. Some example in tempered distributions space are examined.