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


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.


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 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aptekarev, A.I.: Strong asymptotics of polynomials of simultaneous orthogonality for Nikishin systems. Mat. Sb. 190(5), 3–44 (1999). Engl. transl. in Russ. Acad. Sci. Sb. Math. 190, 631–669 (1999) MathSciNetGoogle Scholar
  2. 2.
    Aptekarev, A.I., Kaliaguine, V.A.: Complex rational approximation and difference operators. Rend. Circ. Mat. Palermo, Ser. II, Suppl. 52, 3–21 (1998) MathSciNetGoogle Scholar
  3. 3.
    Aptekarev, A.I., Kaliaguine, V.A., Van Iseghem, J.: The genetic sum’s representation for the moments of a system of Stieltjes functions and its application. Constr. Approx. 16, 487–524 (2000) zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bogoyavlenskii, O.: Integrable dynamical systems associated with the KdV equation. Izv.  Acad.  Nauk SSSR, Ser.  Mat. 51(6) (1987). English transl.: Math.  USSR Izv. 31(3), 435–454 (1988) Google Scholar
  5. 5.
    Bustamente, J., Lopez, G.: Hermite–Padé approximation to a Nikishin type system of analytic functions. Mat. Sb. 183(2), 117–138 (1992). Engl. transl. in Russ. Acad. Sci. Sb. Math. 77, 367–384 (1994) Google Scholar
  6. 6.
    Chihara, T.S., Nevai, P.G.: Orthogonal polynomials and measures with finitely many point masses. J. Approx. Theory 35, 370–380 (1982) zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Dombrowski, J., Nevai, P.: Orthogonal polynomials, measures and recurrence relations. SIAM J. Math. Anal. 17, 752–759 (1986) zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Driver, K., Stahl, H.: Normality in Nikishin systems. Indag. Math. 5(2), 161–187 (1994) zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Driver, K., Stahl, H.: Simultaneous rational approximants to Nikishin systems. I, II. Acta Sci. Math. (Szeged) 60, 245–263 (1995); 61, 261–284 (1995) zbMATHMathSciNetGoogle Scholar
  10. 10.
    Eiermann, M., Varga, R.S.: Zeros and local extreme points of Faber polynomials associated with hypocycloidal domains. Electron. Trans. Numer. Anal. 1, 49–71 (1993) zbMATHMathSciNetGoogle Scholar
  11. 11.
    Gakhov, F.D.: Boundary Value Problems. Gos. Izdat. Fiz.-Mat. Lit., Moscow (1963). English transl. Dover, New York (1990) Google Scholar
  12. 12.
    Geronimo, J.S., Case, K.M.: Scattering theory and polynomials orthogonal on the real line. Trans. Am. Math. Soc. 258, 467–494 (1980) zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Gonchar, A.A., Rakhmanov, E.A., Sorokin, V.N.: On Hermite–Padé approximants for systems of functions of Markov type. Mat. Sb. 188(5), 33–58 (1997). Engl. transl. in Russ. Acad. Sci. Sb. Math. 188(5), 671–696 (1997) MathSciNetGoogle Scholar
  14. 14.
    Graves-Morris, P.R., Saff, E.B.: Vector-valued rational interpolants. In: Graves-Morris, P.R., Saff, E.B., Varga, R.S. (eds.) Rational Approximation and Interpolation. Springer Lecture Notes, vol. 1105, pp. 227–242. Springer, New York (1984) CrossRefGoogle Scholar
  15. 15.
    He, M., Saff, E.B.: The zeros of Faber polynomials for an m-cusped hypocycloid. J. Approx. Theory 78, 410–432 (1994) zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Kac, M., Van Moerbeke, P.: On an explicitly soluble system of nonlinear differential equations related to certain Toda lattices. Adv.  Math. 16, 160–169 (1975) zbMATHCrossRefGoogle Scholar
  17. 17.
    Kaliaguine, V.A.: Hermite–Padé approximants and spectral analysis of nonsymmetric difference operators. Mat. Sb. 185, 79–100 (1994). Engl. transl. in Russ. Acad. Sci. Sb. Math. 82(1), 199–216 (1995) Google Scholar
  18. 18.
    Kaliaguine, V.A.: The operator moment problem, vector continued fractions and an explicit form of the Favard theorem for vector orthogonal polynomials. J. Comput. Appl. Math. 65, 181–193 (1995) zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Mate, A., Nevai, P., Totik, V.: Asymptotics for orthogonal polynomials defined by a recurrence relation. Constr. Approx. 1, 231–248 (1985) zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Moser, J.: Three integrable Hamiltonian systems connected with isospectral deformations. Adv.  Math. 16, 197–220 (1975) zbMATHCrossRefGoogle Scholar
  21. 21.
    Nevai, P.G.: Orthogonal Polynomials. Mem. Am. Math. Soc., vol. 213. Am. Math. Soc., Providence (1979) Google Scholar
  22. 22.
    Nevai, P., Van Assche, W.: Compact perturbations of orthogonal polynomials. Pac. J. Math. 153, 163–184 (1992) zbMATHGoogle Scholar
  23. 23.
    Nikishin, E.M.: On simultaneous Padé approximants. Mat. Sb. 113(155), 499–519 (1980). English transl. in Math. USSR Sb. 41, 409–421 (1980) MathSciNetGoogle Scholar
  24. 24.
    Nikishin, E.M., Sorokin, V.N.: Rational Approximations and Orthogonality. Nauka, Moscow (1988). Transl. Mathem. Monographs, vol. 92. Am. Math. Soc., Providence (1992) zbMATHGoogle Scholar
  25. 25.
    Nuttall, J.: Asymptotics of diagonal Hermite–Padé polynomials. J. Approx. Theory 42(4), 299–386 (1984) zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Sorokin, V.: Integrable nonlinear dynamical systems of Langmuir lattice type. Mat. Zamet. 62, 588–602 (1997). English translation in Math. Notes 62, 488–500 (1997) MathSciNetGoogle Scholar
  27. 27.
    Stieltjes, T.: Recherches sur les fractions continues. Ann. Fac. Sci. Toulouse 8(J), 1–122 (1894); 9(A), 1–47 (1894). Oeuvres Completes, vol. 2, pp. 402–566. Noordhoff, Groningen (1918) MathSciNetGoogle Scholar
  28. 28.
    Van Assche, W.: Asymptotics for orthogonal polynomials and three-term recurrences. In: Nevai, P. (ed.) Orthogonal Polynomials: Theory and Practice, pp. 435–462. Kluwer Academic, Dordrecht (1990) Google Scholar
  29. 29.
    Van Iseghem, J.: Vector orthogonal relations, vector QD-algorithm. J. Comput. Appl. Math. 19, 141–150 (1987) zbMATHMathSciNetGoogle Scholar
  30. 30.
    Wall, H.S.: Analytic Theory of Continued Fractions. Chelsea, New York (1973) Google Scholar

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

Personalised recommendations