# Frames, their relatives and reproducing kernel Hilbert spaces

## TL;DR

This paper explores the relationship between reproducing kernel Hilbert spaces (RKHS) and frame theory, revealing structural insights and limitations in extending Riesz bases to general measure spaces.

## Contribution

It introduces a new RKHS-based approach to analyze frames, proves finite redundancy implies atomic measure spaces, and characterizes the RKHS structure of analysis operator ranges.

## Key findings

- Finite redundancy of continuous frames implies atomic measure spaces.
- All attempts to extend Riesz bases to general measure spaces are limited to discrete cases.
- Range of analysis operators of reproducing pairs can be endowed with an RKHS structure.

## Abstract

This paper considers different facets of the interplay between reproducing kernel Hilbert spaces (RKHS) and stable analysis/synthesis processes: First, we analyze the structure of the reproducing kernel of a RKHS using frames and reproducing pairs. Second, we present a new approach to prove the result that finite redundancy of a continuous frame implies atomic structure of the underlying measure space. Our proof uses the RKHS structure of the range of the analysis operator. This in turn implies that all the attempts to extend the notion of Riesz basis to general measure spaces are fruitless since every such family can be identified with a discrete Riesz basis. Finally, we show how the range of the analysis operators of a reproducing pair can be equipped with a RKHS structure.