Let be introduced the Sobolev-type inner product (f, g) = 1 2 Z 1 −1 f(x)g(x)dx + M [f ′(1)g′(1) + f ′(−1)g′(−1)], where M ≥ 0. In this paper we will prove that for 1 ≤ p ≤ 4 3 there are functions f ∈ L([−1, 1]) whose Fourier expansion in terms of the orthonormal polynomials with respect to the above Sobolev inner product are divergent almost everywhere on [−1, 1]. We also show that, for some v...