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

Let View the MathML source{Qn(α)(x)}n≥0 denote the sequence of monic polynomials orthogonal with respect to the non-discrete Sobolev inner product View the MathML source〈f,g〉=∫−11f(x)g(x)dμ(x)+λ∫−11f′(x)g′(x)dμ(x) Turn MathJax on where View the MathML sourcedμ(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 View the MathML sourceQn(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.