Constructive Approximation

, Volume 16, Issue 4, pp 487–524 | Cite as

The Genetic Sums' Representation for the Moments of a System of Stieltjes Functions and its Application

  • A. Aptekarev
  • V. Kaliaguine
  • J. Van Iseghem


This paper deals with the spectral problems for high-order nonsymmetric difference operators. The method of investigation is based on the analysis of the genetic sums formulas for the moments of the operator. The parameters of these sums are shown to be connected with coefficients of the introduced vector Stieltjes continued fraction. The connections with vector orthogonality, Hermite—Padé approximation, and Hankel determinants are investigated. This gives a tool for the analysis of the solution of the direct and inverse spectral problem of the operator. It is applied to the integration of hierarchy of the discrete KdV equations. The existence of a global solution is proved.

Key words. Genetic sums Stieltjes functions, Nonsymmetric difference operators, Vector orthogonality, Hermite—Padé approximation, Vector continued fraction, Direct and inverse spectral problem, Discrete KdV equations.  AMS Classification. 41A20.21  47A10  47B99  30B70. 


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Copyright information

© Springer-Verlag New York 2000

Authors and Affiliations

  • A. Aptekarev
    • 1
  • V. Kaliaguine
    • 2
  • J. Van Iseghem
    • 3
  1. 1.Keldysh Institute of Applied Mathematics aptekaa@rfbr.ruRU
  2. 2.Nizhny Novgorod State Technical University kalia@waise.nntu.sci-nnov.ruRU
  3. 3.UFR de Mathématiques France jvaniseg.@ano.univ-lille1.frFR

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