# Continuous Donoho-Elad Spark Uncertainty Principle

## TL;DR

This paper extends the concept of spark from finite frames to linear maps on measure spaces, deriving a new uncertainty principle and conditions for sparse solutions in this broader context.

## Contribution

It generalizes the spark notion to measure space linear maps, leading to a novel uncertainty principle and sparse solution criteria.

## Key findings

- Extended spark to measure space linear maps
- Derived a new uncertainty principle for these maps
- Provided conditions for sparse solutions in measure space systems

## Abstract

Donoho and Elad \textit{[Proc. Natl. Acad. Sci. USA, 2003]} introduced the important notion of the spark of a frame, using which they derived a fundamental uncertainty principle. Based on spark, they also provided a necessary and sufficient condition for the uniqueness of sparse solutions to the NP-hard $\ell_0$-minimization problem. In this nano note, we show that the notion of spark can be extended to linear maps whose domains are measure spaces. Using this generalization, we derive an uncertainty principle and provide a sufficient condition for the existence of sparse solutions to linear systems on measure spaces.