On W1,p-convergence of Fourier-Sobolev expansions

Let {qn}n≥0 be the sequence of polynomials orthonormal with respect to the Sobolev inner product 〈f,g〉S:=∫−11f(x)g(x)w0(x)dx+∫−11f′(x)g′(x)w1(x)dx, where w0∈L∞([−1,1]) and w1 is a weight of Kufner-Opic type. We study necessary and/or sufficient conditions for the convergence in the W1,p([−1,1],(w0,w1)) norm of the Fourier expansion in terms of {qn}n≥0, with 1