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On the asymptotics of the ratio of orthogonal polynomials and convergence of multipoint Padé approximants. (English. Russian original) Zbl 0604.30044

Math. USSR, Sb. 56, 207-219 (1987); translation from Mat. Sb., Nov. Ser. 128(170), No. 2, 216-229 (1985).
Let \(d\mu\) be a positive Borel measure on [-1,1] and \(\{\omega_{2n}\}\), \(n\in {\mathbb{N}}^ a \)sequence of polynomials with real coefficients such that \(\omega_{2n}\geq 0\) on [-1,1]. Denote \(L_{n,m}\) the mth monic orthogonal polynomial with respect to the measure \(d\mu /\omega_{2n}\). In these papers the author proves that if the zeros of \(\omega_{2n}\) are bounded away from [-1,1], then \[ (1)\quad \lim_{n}\frac{L_{n,n+k+1}(z)}{L_{n,n+k}(z)}=(z+\sqrt{z^ 2+1})=\phi (z) \] uniformly on each compact subset of \({\mathbb{C}}\setminus [-1,1]\). This result extends a previous well-known theorem of E. A. Rakhmanov [Mat. Sb., Nov. Ser. 103(145), 237-252 (1977; Zbl 0373.30034) and ibid. 118(160), 104-117 (1982; Zbl 0509.30028)]. (1) is used to obtain convergence of multipoint Padé approximants to functions of type \(\mu *(1/z)+r\), where r is a rational function with complex coefficients and poles in \({\mathbb{C}}\setminus [-1,1]\). The paper in Publ. Math. Orsay (reviewed above) is a first draft of the one finally submitted to Mat. Sb. and was published with the consent of this last journal. For the case when (some or all) the zeros of \(\omega_{2n}\) are allowed to tend to [- 1,1] then (1) also holds as long as \(\lim_{n}\sum^{2n}_{k=1}(1- | \phi (x_{n,k})|^{-1})=\infty\) where \((x_{n,k})\), k- 1,...,2n is the set of zeros of \(\omega_{2n}\). This result will appear in Constructive Approximation.

MSC:

30E10 Approximation in the complex plane
41A21 Padé approximation
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
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