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Recently, it has been observed that a sparse trigonometric polynomial, i.e., having only a small number of nonzero coefficients, can be reconstructed exactly from a small number of random samples using basis pursuit (BP) or orthogonal matching pursuit (OMP). In this paper, it is shown that recovery by a BP variant is stable under perturbation of the samples values by noise. A similar partial result for OMP is provided. For BP, in addition, the stability result is extended to (nonsparse) trigonometric polynomials that can be well approximated by sparse ones. The theoretical findings are illustrated by numerical experiments.