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Relative asymptotics for polynomials orthogonal with respect to a discrete Sobolev inner product. (English) Zbl 0840.42017
The asymptotic behavior of polynomials orthogonal with respect to the discrete Sobolev inner product \[ \langle h, g\rangle= \int hg d\mu+ \sum^m_{j= 1} \sum^{N_j}_{i= 1} M_{j, i} h^{(i)}(c_j) g^{(i)} (c_j) \] is investigated. In the definition of the inner product \(\mu\) is a certain type of real or complex measure defined on the real axis, and the \(c_j\) are complex numbers in the complement of \(\text{supp}(\mu)\). The polynomials are called Sobolev orthogonal polynomials. The main group of results deals with relative asymptotics, i.e., the asymptotic behavior \((n\to \infty)\) of the quotient \(Q_n/L_n\), where \(Q_n\) is the Sobolev orthogonal polynomial of order \(n\) and \(L_n\) the polynomial orthogonal with respect to the measure \(\mu\).
Reviewer: H.Stahl (Berlin)

42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
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