Continuous frames and the Kadison-Singer problem

TL;DR

This paper surveys recent advances in continuous frames inspired by the Kadison-Singer problem, including extensions of Lyapunov's theorem and solutions to the discretization problem, highlighting significant progress in frame theory.

Contribution

It presents an extension of Lyapunov's theorem for continuous frames and outlines the solution to the discretization problem, advancing the understanding of frame structures.

Findings

Extension of Lyapunov's theorem for continuous frames

Solution to the discretization problem for frames

Connection to the Kadison-Singer problem

Abstract

In this paper we survey a recent progress on continuous frames inspired by the solution of the Kadison-Singer problem by Marcus, Spielman, and Srivastava. We present an extension of Lyapunov's theorem for discrete frames due to Akemann and Weaver and a similar extension for continuous frames by the author. We also outline a solution of the discretization problem, which was originally posed by Ali, Antoine, and Gazeau, and recently solved by Freeman and Speegle.