Asymptotic properties and Fourier expansions of orthogonal polynomials with a non-discrete Gegenbauer-Sobolev inner product

Let {Q(n)((alpha)) (x)}(n >= 0) denote the sequence of monic polynomials orthogonal with respect to the non-discrete Sobolev inner product < f, g > = integral(1)(-1) f(x)g(x)d mu(x) + lambda integral(1)(-1) f'(x)g'(x)d mu(x) where d mu(x) = (1 - x(2))(alpha-1/2)dx with alpha > -1/2, and lambda > 0. A strong asymptotic on (-1, 1), a Mehler-Heine type formula as well as Sobolev norms of Q(n)((alpha)) are obtained. We also study the necessary conditions for norm convergence and the failure of a.e. convergence of a Fourier expansion in terms of the Sobolev orthogonal polynomials. (C) 2009 Elsevier Inc. All rights reserved.