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 $$\langle f,g\rangle = \int _{-1} f(x)g(x)d\mu(x)+\sum _{k=1} \sum N_k}_{i=0} M_{k,i} f (i)}(a_k)g (i)}(a_k),$$ where the mass points $a_k$ belong to [−1,1], $M_{k,i}\geq 0$, $i = 0,\dots,N_k-1$, 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 Mk,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].