Reproducing pairs of measurable functions

TL;DR

This paper explores the concept of reproducing pairs of weakly measurable functions, generalizing continuous frames, and demonstrates that each pair generates two conjugate dual Hilbert spaces with various examples.

Contribution

It introduces the notion of reproducing pairs of weakly measurable functions and shows they generate dual Hilbert spaces, extending the theory of continuous frames.

Findings

Reproducing pairs generate two conjugate dual Hilbert spaces.

Examples include both discrete and continuous cases.

The concept generalizes continuous frames.

Abstract

We analyze the notion of reproducing pair of weakly measurable functions, which generalizes that of continuous frame. We show, in particular, that each reproducing pair generates two Hilbert spaces, conjugate dual to each other. Several examples, both discrete and continuous, are presented.