  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.
