Advertisement

Construction of Solutions of the Hamburger–Löwner Mixed Interpolation Problem for Nevanlinna Class Functions

  • J. A. Alcober
  • I. M. Tkachenko
  • M. Urrea
Chapter

Abstract

By definition, a Nevanlinna class function \(\varphi \in \Re\) is holomorphic and has a nonnegative imaginary part in the half-plane Im z > 0. In this chapter we also consider Nevanlinna functions which belong to the subclass \(\Re_0 \subset \Re\) such that if \(\varphi(z) \in \Re_0, \lim\nolimits_{z\to \infty} (\varphi(z)/z) = 0\), Im z > 0. Then, due to the Riesz–Herglotz theorem,
$$\varphi(z)=\int\limits^\infty_{-\infty}\frac{d\sigma(t)}{t-z}, \quad {\rm Im} \ z>0,$$
where σ(t) is a nondecreasing function such that
$$\int^\infty_{-\infty} (1+t^2)^{-1} \ d\sigma(t) < \infty$$
. Consider the mixed Löwner–Nevanlinna problem [Löw34, KrNu77, Akh65, KaSt66, CuFi91, CuFi96, AdTk00, UrTkFC01, AdAlTk03], see also [AdTk01(a)] and (for the matrix version of the problem) [AdTk01(b)].

Keywords

Entropy 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [AdTk00]
    Adamyan, V.M., Tkachenko, I.M.: Solution of the truncated Hamburger moment problem according to M.G. Krein. Operator Theory Adv. Appl., OT-118, 33-51 (2000).MathSciNetGoogle Scholar
  2. [AdTk01(a)]
    Adamyan, V.M., Tkachenko, I.M.: Truncated Hamburger moment problems with constraints. Math. Studies, 189, 321-333 (2001).MathSciNetGoogle Scholar
  3. [AdTk01(b)]
    Adamyan, V.M., Tkachenko, I.M.: Truncated Hamburger matrix moment problems with constraints. Proc. Appl. Math. Mech., 1, 420-421 (2001).CrossRefGoogle Scholar
  4. [AdAlTk03]
    Adamyan, V.M., Alcober, J.A., Tkachenko, I.M.: Reconstruction of distributions by their moments and local constraints. Appl. Math. Research eXpress, 2003, 33-70.Google Scholar
  5. [Akh65]
    Akhiezer, N.I.: The Classical Moment Problem and Some Related Questions in Analysis, Hafner, New York (1965).MATHGoogle Scholar
  6. [CuFi91]
    Curto, R.E., Fialkow, L.A.: Recursiveness, positivity, and truncated moment problems. Houston J. Math., 17, 603-635 (1991).MATHMathSciNetGoogle Scholar
  7. [CuFi96]
    Curto, R.E., Fialkow, L.A.: Solutions of the truncated moment problem for flat data. Mem. Amer. Math. Soc., 119 (1996).Google Scholar
  8. [DeDy81]
    Dewilde, P., Dym, H.: Schur recursions, error formulas, and convergence of rational estimators for stationary stochastic sequences. IEEE Trans. Information Theory, IT-27, 446-461 (1981).CrossRefMathSciNetGoogle Scholar
  9. [KrNu77]
    Krein, M.G., Nudel'man, A.A.: The Markov Moment Problem and Extremal Problems, American Mathematical Society, Providence, RI (1977).MATHGoogle Scholar
  10. [KaSt66]
    Karlin, S., Studden, W.S.: Tschebyscheff Systems with Applications in Analysis and Statistics, Interscience, New York (1966).Google Scholar
  11. [KhaTa85]
    Khargonekar, P., Tannenbaum, A.: Non-Euclidian metrics and the robust stabilization of systems with parameter uncertainty. IEEE Trans. Automat. Contr., AC-30, 1005-1013 (1985).CrossRefMathSciNetGoogle Scholar
  12. [Löw34]
    Löwner, K.: Über monotone Matrixfunktionen. Math. Z., 38, 177 (1934).CrossRefMathSciNetGoogle Scholar
  13. [TkUr99]
    Tkachenko, I.M., Urrea, M.: Determination of the adjustable moment model parameter by the minimization of the Shannon entropy. Z. Angew. Math. Mech., 79, 789-790 (1999).Google Scholar
  14. [UrTkFC01]
    Urrea, M., Tkachenko, I.M., Fernández de Córdoba, P.: The Nevanlinna theorem of the classical theory of moments revisited. J. Appl. Anal., 7, 209-224 (2001).MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Birkhäuser Boston 2010

Authors and Affiliations

  1. 1.Universidad Politécnica de ValenciaValenciaSpain

Personalised recommendations