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On symmetric differential operators associated with Sobolev orthogonal polynomials: A characterization. (English) Zbl 0980.42018
Authors’ abstract: “Given the Sobolev bilinear form \[ (f,g)_S=\langle u_0, fg\rangle + \langle u_1, f'g'\rangle , \] with \(u_0\) and \(u_1\) linear functionals, a characterization of the linear second-order differential operators with polynomial coefficients, symmetric with respect to \(( \cdot , \cdot)_S\) in terms of \(u_0\) and \(u_1\) is obtained. In particular, several interesting functionals \(u_0\) and \(u_1\) are considered, recovering as particular cases of our study, results already known in the literature”.
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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