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

Error localization of best L1 polynomial approximants