We consider some discrete approximation polynomials, namely discrete de la Vallée Poussin means, which have been recently deduced from certain delayed arithmetic means of the Fourier–Jacobi partial sums, in order to get a near–best approximation in suitable spaces of continuous functions equipped with the weighted uniform norm. By the present paper we aim to analyze the behavior of such discrete de la Vallée means in weighted L 1 spaces, where we provide error bounds for several classes of functions, included functions of bounded variation. In all the cases, under simple conditions on the involved Jacobi weights, we get the best approximation order. During our investigations, a weighted L 1 Marcinkiewicz type inequality has been also stated.