The uniform convergence of Fourier series in a system of polynomials orthogonal in the sense of Sobolev and associated to Jacobi polynomials

We establish that the Fourier series in the Sobolev system of polynomials ${\mathcal P}_r^{\alpha,\beta}$, with $-1 < \alpha,\beta \le 0$, associated to the Jacobi polynomials converge uniformly on $[-1,1]$ to functions in the Sobolev space $W^r_{L^1_{\rho(\alpha,\beta)}}$, where $\rho(\alpha,\beta)$ is the Jacobi weight. We show also that the Fourier series converges in the norm of the Sobolev space $W^r_{L^p_{\rho(A,B)}}$ with $p>1$ under the Muckenhoupt conditions.