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Let dmualfa,beta(x)=(1−x)alfa(1+x)betadx,alfa,beta>−1, be the Jacobi measure supported on the interval [−1,1]. Let us introduce the Sobolev inner product ⟨f,g⟩S=∑Nj=0lambdaj∫1−1f(j)(x)g(j)(x)dmualfa,beta(x), where lambdaj≥0 for 0≤j≤N−1 and lambdaN>0. In this paper we obtain some asymptotic results for the sequence of orthogonal polynomials with respect to the above Sobolev inner product. Furthermore, we prove a Cohen type inequality for Fourier expansions in terms of such polynomials.