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On Fourier series of a discrete Jacobi–Sobolev inner product. (English) Zbl 1019.42014
Summary: Let \(\mu\) be the Jacobi measure supported on the interval \([-1,1]\) 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 \(a_k\), \(1\leq k\leq K\), are real numbers such that \(|a_k|> 1\) and \(M_{k,i}> 0\) for all \(k\), \(i\). This paper is a continuation of [F. Marcellán, B. P. Osilenker and I. A. Rocha, “On Fourier series of Jacobi-Sobolev orthogonal polynomials”, J. Inequal. Appl. 7, 673-699 (2002; Zbl 1016.42014)] 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 \((-1,1)\bigcup^K_{k=1}\{a_k\}\), and the derivatives of the series converge to \(f^{(i)}(a_k)\) 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 \(\mu\). No additional conditions are needed.

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