# Error localization of best L1 polynomial approximants

## TL;DR

This paper establishes a connection between best L0 and L1 polynomial approximants for corrupted polynomials, demonstrating an error localization property and proposing an improved approximation algorithm.

## Contribution

It introduces a continuous analogue of compressed sensing principles for polynomial approximation and develops an enhanced method for computing best L1 polynomial approximants.

## Key findings

- Best L0 and L1 polynomial approximants are nearly equal for corrupted polynomials.
- Error localization property of best L1 polynomial approximants is demonstrated.
- An improved algorithm for computing best L1 polynomial approximants is proposed.

## Abstract

An important observation in compressed sensing is that the $\ell_0$ minimizer of an underdetermined linear system is equal to the $\ell_1$ minimizer when there exists a sparse solution vector and a certain restricted isometry property holds. Here, we develop a continuous analogue of this observation and show that the best $L_0$ and $L_1$ polynomial approximants of a polynomial that is corrupted on a set of small measure are nearly equal. We go on to demonstrate an error localization property of best $L_1$ polynomial approximants and use our observations to develop an improved algorithm for computing best $L_1$ polynomial approximants to continuous functions.