On -convergence of Fourier–Sobolev expansions

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