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Russian Mathematics

, Volume 63, Issue 6, pp 58–73 | Cite as

Resolvent Matrix of the Truncated Nevanlinna-Pick Matrix Interpolation Problem via Orthogonal Rational Functions in the Stieltjes Class

  • A. E. Choque-RiveroEmail author
Article
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Abstract

A new representation of the resolvent matrix and extremal solutions of the truncated Nevanlinna-Pick matrix interpolation problem in the Stieltjes class functions via orthogonal matrix rational functions on [0, ∞) is given.

Key words

truncated Nevanlinna-Pick matrix interpolation problem matrix-valued orthogonal rational functions resolvent matrix 

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Notes

Acknowledgments

The author thanks the referee for helpful comments and suggestions.

Funding

The author was partially supported by SNI Conacyt, CIC-UMSNH and PROMEP México.

References

  1. 1.
    Akhiezer, N.I., Glazman, I.M. Theory of linear operators in Hilbert space (Dover Publications Inc., New York, 1993).zbMATHGoogle Scholar
  2. 2.
    Blomqvist, A., Lindquist, A., Nagamune, R. “Matrix-valued Nevanlinna-Pick interpolation with complexity constraint: an optimization approach”, IEEE Trans. Automat. Contr. 48 (12), 2172–2190 (2003).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bultheel, A., De Moor, A. “Rational approximation in linear systems and control”, J. Comput. Appl. Math. 121 (1–2), 355–378 (2000).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Choque Rivero, A.E. “On Dyukarev’s resolvent matrix for a truncated Stieltjes matrix moment problem under the view of orthogonal matrix polynomials”, Linear Algebra Appl. 474, 44–109 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Choque Rivero, A.E. “On matrix Hurwitz type polynomials and their interrelations to Stieltjes positive definite sequences and orthogonal matrix polynomials”, Linear Algebra Appl. 476, 56–84 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Choque Rivero, A.E. “On the solution set of the admissible control problem via orthogonal polynomials”, IEEE Trans. Autom. Control 62 (10), 5213–5219 (2017).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Choque-Rivero, A.E., “The Kharitonov theorem and robust stabilization via orthogonal polynomials”, Visn. Khark. Univ., Ser. Mat. Prykl. Mat. Mekh. 86, 49–68 (2017).zbMATHGoogle Scholar
  8. 8.
    Choque-Rivero A.E. “Hurwitz polynomials and orthogonal polynomials generated by Routh-Markov parameters”, Mediterr. J. Math, https://doi.org/10.1007/s00009-018-1083-2 (2018).Google Scholar
  9. 9.
    Delsarte, P., Genin, Y., Kamp, Y. “On the role of the Nevanlinna-Pick problem in circuit and system theory”, Int. J. Circ. Theor. and App. 9 (2), 177–187 (1981).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Derevyagin, M. “The Jacobi matrices approach to Nevanlinna-Pick problems”, J. Approx. Theory 163, 117–142 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Derevyagin, M., Zhedanov, A.S. “An operator approach to multipoint Padé approximations”, J. Approx. Theory 157, 70–88 (2019).CrossRefzbMATHGoogle Scholar
  12. 12.
    Dyukarev, Yu.M. “The multiplicative structure of resolvent matrices of interpolation problems in the Stieltjes class”, Vestnik Kharkov Univ. Ser. Mat. Prikl. Mat. i Mekh. 458, 143–153 (1999).zbMATHGoogle Scholar
  13. 13.
    Dyukarev, Yu.M. “Nevanlinna-Pick Problem for Stieltjes Matrix Functions”, Ukrainian Mathematical Journal 56 (3), 446–465 (2004).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Dyukarev, Yu.M. “Indeterminacy criteria for the Stieltjes matrix moment problem”, Mathematical Notes 75 (1–2), 66–82 (2004).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Dyukarev, Yu.M. “Indeterminacy of interpolation problems in the Stieltjes class”, Mat. Sb. 196 (3), 61–88 (2005).MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Dyukarev, Yu.M. “A generalized Stieltjes criterion for the complete indeterminacy of interpolation problems”, Math. Notes 84 (1), 23–39 (2008).MathSciNetzbMATHGoogle Scholar
  17. 17.
    Dyukarev, Yu.M. “Criterion for complete indeterminacy of limiting interpolation problem of Stieltjes type in terms of orthonormal matrix functions”, Russian Math. 59 (4), 61–88 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Dyukarev, Yu.M. “Geometric and operator measures of degeneracy for the set of solutions to the Stieltjes matrix moment problem”, Mat. Sb. 207 (4), 47–64 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Dyukarev, Yu.M., Katsnel’son, V.E. “Multiplicative and additive Stieltjes classes of analytic matrixvalued functions and interpolation problems connected with them. I.”, Teor. Funktsii Funktsional. Anal. i Prilozhen. 36, 13–27 (1981) [in Russian].zbMATHGoogle Scholar
  20. 20.
    Dyukarev, Yu.M., Serikova, I.Yu. “Step-by-step solving of ordered interpolational problem for Stieltjes functions”, Russian Math. 61 (6), 13–26 (2017).MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Kimura H. “State space approach to the classical interpolation problem and its applications” (in: H. Nijmeijier and J.M. Schumacher (Eds.), Three decades of mathematical system theory, Springer Verlag, Berlin, 243–275 (1989)).CrossRefGoogle Scholar
  22. 22.
    Krein, M.G., Nudel’man, A.A. The Markov moment problem and extremal problems (Translations of Mathematical Monographs, AMS 50, Providence, 1977).CrossRefzbMATHGoogle Scholar

Copyright information

© Allerton Press, Inc. 2019

Authors and Affiliations

  1. 1.Instituto de Física y MatemáticasUniversidad Michoacana de San Nicolás de Hidalgo CUMoreliaMexico

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