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On a class of Sobolev scalar products in the polynomials. (English) Zbl 1042.42018

This paper discusses some asymptotic properties as well as the location of zeros of orthogonal polynomials with respect to the Sobolev inner product \[ \langle f,g \rangle=\int_{\Omega_0} p(z)\overline{q(z)}\,d\mu_0 + \int_{\Omega_1} p'(z)\overline{q'(z)}\,d\mu_1, \] where \(\mu_0\) and \(\mu_1\) are positive Borel measures in the complex plane with supports \(\Omega_0\) and \(\Omega_1\), respectively. In fact, under the following condition on the corresponding \(L^2\) norms \(\|p\|_{\mu_1}\leq M\|p\|_S\), where \(M\) is a positive constant and \(\|p\|_S\) is the norm induced by the Sobolev inner product, and using the connection between orthogonal polynomials and Hessenberg matrices the authors obtain several results on the location of the zeros or the polynomials orthogonal with respect to the aforesaid Sobolev inner product as well as some Markov-type asymptotics. Finally, they obtain several operator-theoretic results such as the determinacy of the associated Sobolev moment problem and the completeness of the polynomials in a certain weighted Sobolev space.

MSC:

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.)
42C30 Completeness of sets of functions in nontrigonometric harmonic analysis
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