Relative asymptotics and Fourier series of orthogonal polynomials with a discrete Sobolev inner product

Abstract

Let μ be a finite positive Borel measure supported in [−1,1] and introduce the discrete Sobolev-type inner product 〈 f,g〉= ∫ −1 1 f(x)g(x) dμ(x)+ ∑ k=1 K ∑ i=0 N k M k,i f (i)(a k) g (i)(a k), where the mass points a k belong to [−1,1], M k,i⩾0, i=0,…,N k−1 , and M k, N k >0. In this paper, we study the asymptotics of the Sobolev orthogonal polynomials by comparison with the orthogonal polynomials with respect to the measure μ and we prove that they have the same asymptotic behaviour. We also study the pointwise convergence of the Fourier series associated to this inner product provided that μ is the Jacobi measure. We generalize the work done by F. Marcellán and W. Van Assche where they studied the asymptotics for only one mass point in [−1,1]. The same problem with a finite number of mass points off [−1,1] was solved by G. López, F. Marcellán and W. Van Assche in a more general setting: they consider the constants M k, i to be complex numbers. As regards the Fourier series, we continue the results achieved by F. Marcellán, B. Osilenker and I.A. Rocha for the Jacobi measure and mass points in R⧹[−1,1] .