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

Abstract

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