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Relative asymptotics and Fourier series of orthogonal polynomials with a discrete Sobolev inner product. (English) Zbl 1014.42019
Summary: Let \(\mu\) be a finite positive Borel measure supported in \([-1,1]\) and introduce the discrete Sobolev-type inner product \[ \langle f,g\rangle = \int^1_{-1} f(x)g(x)d\mu(x)+\sum^K_{k=1} \sum^{N_k}_{i=0} M_{k,i} f^{(i)}(a_k)g^{(i)}(a_k), \] 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 \(\mu\) 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 \(\mu\) 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 considered 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 \(\mathbb{R}\setminus [-1,1]\).

MSC:
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
33C47 Other special orthogonal polynomials and functions
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