Advertisement

On Canonical Solutions of a Moment Problem for Rational Matrix-valued Functions

  • Bernd FritzscheEmail author
  • Bernd Kirstein
  • Andreas Lasarow
Conference paper
Part of the Operator Theory: Advances and Applications book series (OT, volume 221)

Abstract

We discuss extremal solutions of a certain finite moment problem for rational matrix functions which satisfy an additional rank condition.W e will see, among other things, that these solutions are molecular nonnegative Hermitian matrix-valued Borel measures on the unit circle and that these measures have a particular structure.W e study the above-mentioned solutions in all generality, but later focus on the nondegenerate case.In this case, the family of these special solutions can be parametrized by the set of unitary matrices.T his realization allows us to further examine the structure of these solutions.H ere, the analysis of the structural properties relies, to a great extent, on the theory of orthogonal rational matrix functions on the unit circle.

Keywords

Nonnegative Hermitian matrix measures matrix moment problem orthogonal rational matrix functions matricial Carathéodory functions. 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aronszajn, N.: Theory of reproducing kernels, Trans. Amer. Math. Soc. 68 (1950), 337-404.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Arov, D.Z.: Regular J-inner matrix-functions and related continuation problems, in: Linear Operators in Function Spaces, Operator Theory: Adv. Appl. Vol. 43, Birkhäuser, Basel, 1990, pp. 63-87.Google Scholar
  3. 3.
    Arov, D.Z.; Dym, H.: J-Contractive Matrix Valued Functions and Related Topics, Encyclopedia Math. and its Appl. 116, Cambridge University Press, Cambridge 2008.CrossRefGoogle Scholar
  4. 4.
    Arov, D.Z.; Krein, M.G.: The problem of finding the minimum entropy in indeterminate problems of continuation (Russian), Funct. Anal. Appl. 15 (1981), 61-64.MathSciNetGoogle Scholar
  5. 5.
    Ben-Artzi, A.; Gohberg, I.: Orthogonal polynomials over Hilbert modules, in: Non-selfadjoint Operators and Related Topics, Operator Theory: Adv. Appl. Vol. 73, Birkhäuser, Basel, 1994, pp. 96-126.Google Scholar
  6. 6.
    Bultheel, A.: Inequalities in Hilbert modules of matrix-valued functions, Proc. Amer. Math. Soc. 85 (1982), 369-372.MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Bultheel A.; Cantero, M.J.: A matricial computation of rational quadrature formulas on the unit circle, Numer. Algorithms 52 (2009), 47-68.MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Bultheel, A.; González-Vera, P.; Hendriksen, E.; Njästad, O.: Orthogonal rational functions and quadrature on the unit circle, Numer. Algorithms 3 (1992), 105-116.MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Bultheel, A.; González-Vera, P.; Hendriksen, E.; Njästad, O.: Moment problems and orthogonal functions, J. Comput. Appl. Math. 48 (1993), 49-68.MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Bultheel, A.; González-Vera, P.; Hendriksen, E.; Njästad, O.: Quadrature formulas on the unit circle based on rational functions, J. Comput. Appl. Math. 50 (1994), 159-170.MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Bultheel, A.; González-Vera, P.; Hendriksen, E.; Njästad, O.: A rational moment problem on the unit circle, Methods Appl. Anal. 4 (1997), 283-310.MathSciNetzbMATHGoogle Scholar
  12. 12.
    Bultheel, A.; González-Vera, P.; Hendriksen, E.; Njästad, O.: Orthogonal Rational Functions, Cambridge Monographs on Applied and Comput. Math. 5, Cambridge University Press, Cambridge 1999.Google Scholar
  13. 13.
    Cantero, M.J.; Cruz-Barroso, R.; Gonzälez-Vera, P.: A matrix approach to the computation of quadrature formulas on the unit circle, Appl. Numer. Math. 58 (2008), 296-318.MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Choque Rivero, A.E.; Lasarow, A.; Rahn, A.: On ranges and Moore-Penrose inverses related to matrix Carathéodory and Schur functions, Complex Anal. Oper. Theory 5 (2011), 513-543.MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    de la Calle Ysern, B.: Error bounds for rational quadrature formulae of analytic functions, Numer. Math. 101 (2005), 251-271.MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    de la Calle Ysern, B.; González-Vera, P.: Rational quadrature formulae on the unit circle with arbitrary poles, Numer. Math. 107 (2007), 559-587.MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Delsarte, P.; Genin, Y.; Kamp, Y.: Orthogonal polynomial matrices on the unit circle, IEEE Trans. Circuits and Systems CAS-25 (1978), 149-160.Google Scholar
  18. 18.
    Dubovoj, V.K.; Fritzsche, B.; Kirstein, B.: Matricial Version of the Classical Schur Problem, Teubner-Texte zur Mathematik 129, B.G. Teubner, Stuttgart-Leipzig 1992.Google Scholar
  19. 19.
    Dym, H.: J Contractive Matrix Functions, Reproducing Kernel Hilbert Spaces and Interpolation, CBMS Regional Conf. Ser. Math. 71, Providence, R.I. 1989.Google Scholar
  20. 20.
    Ellis, R.L.; Gohberg, I.: Extensions of matrix-valued inner products on modules and the inversion formula for block Toeplitz matrices, in: Operator Theory and Analysis, Operator Theory: Adv. Appl. Vol. 122, Birkläuser, Basel, 2001, pp. 191-227.Google Scholar
  21. 21.
    Ellis, R.L.; Gohberg, I.; Lay, D.C.: Infinite analogues of block Toeplitz matrices and related orthogonal functions, Integral Equations Operator Theory 22 (1995), 375-419.MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Fritzsche, B.; Fuchs, S.; Kirstein, B.: A Schur type matrix extension problem V, Math. Nachr. 158 (1992), 133-159.MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Fritzsche, B.; Kirstein, B.; Lasarow, A.: On a moment problem for rational matrix-valued functions, Linear Algebra Appl. 372 (2003), 1-31.MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Fritzsche, B.; Kirstein, B.; Lasarow, A.: On rank invariance of moment matrices of nonnegative Hermitian-valued Borel measures on the unit circle, Math. Nachr. 263/264 (2004), 103-132.MathSciNetCrossRefGoogle Scholar
  25. 25.
    Fritzsche, B.; Kirstein, B.; Lasarow, A.: On Hilbert modules of rational matrix-valued functions and related inverse problems, J. Comput. Appl. Math. 179 (2005), 215-248.MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Fritzsche, B.; Kirstein, B.; Lasarow, A.: Orthogonal rational matrix-valued functions on the unit circle, Math. Nachr. 278 (2005), 525-553.MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Fritzsche, B.; Kirstein, B.; Lasarow, A.: Orthogonal rational matrix-valued functions on the unit circle: Recurrence relations and a Favard-type theorem, Math. Nachr. 279 (2006), 513-542.MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Fritzsche, B.; Kirstein, B.; Lasarow, A.: The matricial Carathéodory problem in both nondegenerate and degenerate cases, in: Interpolation, Schur Functions and Moment Problems, Operator Theory: Adv. Appl. Vol. 165, Birkhäuser, Basel, 2006, pp. 251-290.Google Scholar
  29. 29.
    Fritzsche, B.; Kirstein, B.; Lasarow, A.: On a class of extremal solutions of the nondegenerate matricial Carathéodory problem, Analysis 27 (2007), 109-164.MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Fritzsche, B.; Kirstein, B.; Lasarow, A.: On a class of extremal solutions of a moment problem for rational matrix-valued functions in the nondegenerate case I, Math. Nachr. 283 (2010), 1706-1735.MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Fritzsche, B.; Kirstein, B.; Lasarow, A.: On a class of extremal solutions of a moment problem for rational matrix-valued functions in the nondegenerate case II, J. Comput. Appl. Math. 235 (2010), 1008-1041.MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Fritzsche, B.; Kirstein, B.; Lasarow, A.: Para-orthogonal rational matrix-valued functions on the unit circle, Oper. Matrices (in press), OaM-0412.Google Scholar
  33. 33.
    Gautschi, W.: A survey of Gauss-Christoffel quadrature formulae, E.B. Christoffel, The Influence of His Work on Mathematical and Physical Sciences, Birkhäuser, Basel, 1981, pp. 72-147.Google Scholar
  34. 34.
    Gautschi, W.; Gori, L.; Lo Cascio, M.L.: Quadrature rules for rational functions, Numer. Math. 86 (2000), 617-633.MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Geronimus, Ja.L.: Polynomials orthogonal on a circle and their applications (Russian), Zap. Naučno-Issled. Inst. Mat. Meh. Har’kov. Mat. Obšč. 19 (1948), 35-120.MathSciNetGoogle Scholar
  36. 36.
    Itoh, S.: Reproducing kernels in modules over C*-algebras and their applications, Bull. Kyushu Inst. Tech. Math. Natur. Sci. 37 (1990), 1-20.zbMATHGoogle Scholar
  37. 37.
    Jones, W.B.; Njästad, O.; Thron, W.J.: Moment theory, orthogonal polynomials, quadrature, and continued fractions associated with the unit circle, Bull. London Math. Soc. 21 (1989), 113-152.MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Kats, I.S.: On Hilbert spaces generated by monotone Hermitian matrix-functions (Russian), Zap. Mat. Otd. Fiz.-Mat. Fak. i Har’kov. Mat. Obšč. 22 (1950), 95-113.Google Scholar
  39. 39.
    Lasarow, A.: Dual pairs of orthogonal systems of rational matrix-valued functions on the unit circle, Analysis 26 (2006), 209-244.MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Lasarow, A.: More on a class of extremal solutions of a moment problem for rational matrix-valued functions in the nondegenerate case, J. Approx. Theory 163 (2011), 864-887.MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Rosenberg, M.: The square integrability of matrix-valued functions with respect to a non-negative Hermitian measure, Duke Math. J. 31 (1964), 291-298.MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Rosenberg, M.: Operators as spectral integrals of operator-valued functions from the study of multivariate stationary stochastic processes, J. Mult. Anal.4 (1974), 166-209.zbMATHCrossRefGoogle Scholar
  43. 43.
    Rosenberg, M.: Spectral integrals of operator-valued functions - II. From the study of stationary processes, J. Mult. Anal.6 (1976), 538-571.zbMATHCrossRefGoogle Scholar
  44. 44.
    Sakhnovich, A.L.: On a class of extremal problems (Russian), Izv. Akad. Nauk SSSR, Ser. Mat. 51 (1987), 436-443.zbMATHGoogle Scholar
  45. 45.
    Velázquez, L.: Spectral methods for orthogonal rational functions, J. Funct. Anal. 254 (2008), 954-986.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel 2012

Authors and Affiliations

  • Bernd Fritzsche
    • 1
    Email author
  • Bernd Kirstein
    • 1
  • Andreas Lasarow
    • 2
  1. 1.Fakultät für Mathematik und InformatikUniversität LeipzigLeipzigGermany
  2. 2.Departement ComputerwetenschappenKatholieke Universiteit LeuvenLeuvenBelgium

Personalised recommendations