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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.

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