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Constructive Approximation

, Volume 30, Issue 2, pp 175–223 | Cite as

Higher-Order Three-Term Recurrences and Asymptotics of Multiple Orthogonal Polynomials

  • A. I. Aptekarev
  • V. A. Kalyagin
  • E. B. Saff
Article

Abstract

The asymptotic theory is developed for polynomial sequences that are generated by the three-term higher-order recurrence
$$Q_{n+1}=zQ_{n}-a_{n-p+1}Q_{n-p},\quad p\in\mathbb{N},n\geq p,$$
where z is a complex variable and the coefficients a k are positive and satisfy the perturbation condition ∑ n=1 |a n a|<∞. Our results generalize known results for p=1, that is, for orthogonal polynomial sequences on the real line that belong to the Blumenthal–Nevai class. As is known, for p≥2, the role of the interval is replaced by a starlike set S of p+1 rays emanating from the origin on which the Q n satisfy a multiple orthogonality condition involving p measures. Here we obtain strong asymptotics for the Q n in the complex plane outside the common support of these measures as well as on the (finite) open rays of their support. In so doing, we obtain an extension of Weyl’s famous theorem dealing with compact perturbations of bounded self-adjoint operators. Furthermore, we derive generalizations of the classical Szegő functions, and we show that there is an underlying Nikishin system hierarchy for the orthogonality measures that is related to the Weyl functions. Our results also have application to Hermite–Padé approximants as well as to vector continued fractions.

Keywords

Higher-order recurrences Polynomials Multiple orthogonality Hermite–Padé approximants Vector continued fractions Faber polynomials Spectral measures Difference operators Weyl’s theorem Nikishin systems Szegő function 

Mathematics Subject Classification (2000)

41A20 41A21 41A10 47B99 30B70 

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Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • A. I. Aptekarev
    • 1
  • V. A. Kalyagin
    • 2
  • E. B. Saff
    • 3
  1. 1.Keldysh Institute of Applied MathematicsRussian Academy of SciencesMoscowRussia
  2. 2.State University-Higher School of EconomicsNizhny NovgorodRussia
  3. 3.Center for Constructive Approximation, Department of MathematicsVanderbilt UniversityNashvilleUSA

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