Hermite interpolation and Sobolev orthogonality

In this paper, we study orthogonal polynomials with respect to the bilinear form (f, g)(S) = V(f)AV(g)(T) + [u, f((N)) g((N))], where V(f) = (f(c(0)), f '(c(0)), ..., f((n0-1))(c(0)), ..., f(c(p)), f '(c(p)), ..., f((np)-(1))(c(p))), u is a regular linear functional on the linear space P of real polynomials, c(0), c(1), ..., c(p) are distinct real numbers, n(0), n(1), ..., n(p) are positive integer numbers, N = n(0) + n(1) + ... + n(p), and A is a N x N real matrix with all its principal submatrices nonsingular. We establish relations with the theory of interpolation and approximation.