On Fourier series of a discrete Jacobi-Sobolev inner product

  • Authors:

  • F. Marcellán;B. P. Osilenker;I. A. Rocha

  • Affiliations:

  • Departamento de Matemáticas, Universidad Carlos III de Madrid, Avda. Universidad 20, Leganés, Madrid, Spain;Department of Mathematics, Moscow State Civil Engineering University, Moscow, Russia;Departamento de Mathemática Aplicada, E.U.I.T. Telecommunicación, Universidad Politécnica de Madrid, Ctra. de Valencia Km. 7, Madrid, Spain

  • Venue:
  • Journal of Approximation Theory
  • Year:
  • 2002

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Abstract

Let µ be the Jacobi measure supported on the interval [-1, 1] and introduce the discrete Sobolev-type inner product f,g-11 f(x)g(x)dµ(x) + Σk=1k Σi=0NkMk,if(i)(ak)g(i)(ak), where ak, 1 ≤ k ≤ K, are real numbers such that |ak| Mk,i k, i. This paper is a continuation of Marcellan et al. (On Fourier series of Jacobi-Sobolev orthogonal polynomials, J. Inequal. Appl., to appear) and our main purpose is to study the behaviour of the Fourier series associated with such a Sobolev inner product. For an appropriate function f, we prove here that the Fourier-Sobolev series converges to f on (-1,1)∪k=1K{ak}, and the derivatives of the series converge to f(i)(ak) for all i and k. Roughly speaking, the term appropriate means here the same as we need for a function f in order to have convergence for its Fourier series associated with the standard inner product given by the measure µ. No additional conditions are needed.