Reproducing pairs of measurable functions and partial inner product spaces

TL;DR

This paper extends the theory of reproducing pairs of weakly measurable functions by exploring cases where these functions take values in partial inner product spaces, providing new examples and generalizations.

Contribution

It introduces the analysis of reproducing pairs within the framework of partial inner product spaces, broadening the scope beyond continuous frames.

Findings

Several discrete and continuous examples are presented.

The generalization to PIP spaces offers new insights into reproducing pairs.

The work extends the mathematical framework of weakly measurable functions.

Abstract

We continue the analysis of reproducing pairs of weakly measurable functions, which generalize continuous frames. More precisely, we examine the case where the defining measurable functions take their values in a partial inner product space (PIP spaces). Several examples, both discrete and continuous, are presented.