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

Let {Qn(α)(x)}n≥0 denote the sequence of monic polynomials orthogonal with respect to the non-discrete Sobolev inner product 〈f,g〉=∫−11f(x)g(x)dμ(x)+λ∫−11f′(x)g′(x)dμ(x) where dμ(x)=(1−x2)α−1/2dx with α>−1/2α>−1/2, and λ>0λ>0. A strong asymptotic on (−1,1)(−1,1), a Mehler–Heine type formula as well as Sobolev norms of Qn(α) 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.