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Journal d'Analyse Mathématique

, Volume 135, Issue 1, pp 271–312 | Cite as

Perturbation of chains of de Branges spaces

  • Harald Woracek
Article
  • 17 Downloads

Abstract

We investigate the structure of the set of de Branges spaces of entire functions which are contained in a space L2(μ). Thereby, we follow a perturbation approach. The main result is a growth dependent stability theorem. Namely, assume that measures μ1 and μ2 are close to each other in a sense quantified relative to a proximate order. Consider the sections of corresponding chains of de Branges spaces C1 and C2 which consist of those spaces whose elements have finite (possibly zero) type with respect to the given proximate order. Then either these sections coincide or one is smaller than the other but its complement consists of only a (finite or infinite) sequence of spaces.

Among other situations, we apply—and refine—this general theorem in two important particular situations
  1. (1)

    the measures μ1 and μ2 differ in essence only on a compact set; then stability of whole chains rather than sections can be shown

     
  2. (2)

    the linear space of all polynomials is dense in L2(μ2); then conditions for density of polynomials in the space L2(μ2) are obtained.

     
In the proof of the main result, we employ a method used by P. Yuditskii in the context of density of polynomials. Another vital tool is the notion of the index of a chain, which is a generalisation of the index of determinacy of a measure having all power moments. We undertake a systematic study of this index, which is also of interest on its own right.

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References

  1. [Bar06]
    A. D. Baranov, Completeness and Riesz bases of reproducing kernels in model subspaces, Int. Math. Res. Not. (2006), Art. ID 81530, 34.Google Scholar
  2. [BBH07]
    A. D. Baranov, A. Borichev, and V. Havin, Majorants of meromorphic functions with fixed poles, Indiana Univ. Math. J. 56 (2007), 1595–1628.MathSciNetCrossRefMATHGoogle Scholar
  3. [BD95]
    C. Berg and A. J. Duran, The index of determinacy for measures and the l2-norm of orthonormal polynomials, Trans. Amer. Math. Soc. 347 (1995), 2795–2811.MathSciNetMATHGoogle Scholar
  4. [BP07]
    C. Berg and H. L. Pedersen, Logarithmic order and type of indeterminate moment problems, Difference Equations, Special Functions and Orthogonal Polynomials (with an appendix by Walter Hayman), World Sci. Publ., Hackensack, NJ, 2007, pp. 51–79.CrossRefGoogle Scholar
  5. [BS11]
    A. Borichev and M. Sodin, Weighted exponential approximation and non-classical orthogonal spectral measures, Adv. Math. 226 (2011), 2503–2545.MathSciNetCrossRefMATHGoogle Scholar
  6. [Bra59a]
    L. de Branges, Some Hilbert Spaces of Entire Functions, Proc. Amer. Math. Soc. 10 (1959), 840–846.MathSciNetCrossRefMATHGoogle Scholar
  7. [Bra59b]
    L. de Branges, Some mean squares of entire functions, Proc. Amer. Math. Soc. 10 (1959), 833–839.MathSciNetCrossRefMATHGoogle Scholar
  8. [Bra61]
    L. de Branges, Some Hilbert spaces of entire functions. II, Trans. Amer. Math. Soc. 99 (1961), 118–152.MathSciNetCrossRefMATHGoogle Scholar
  9. [Bra68]
    L. de Branges, Hilbert Spaces of Entire Functions, Prentice-Hall Inc., Englewood Cliffs, NJ, 1968.MATHGoogle Scholar
  10. [Dym70]
    H. Dym. An introduction to de Branges spaces of entire functions with applications to differential equations of the Sturm-Liouville type, Advances in Math. 5 (1970), 395–471.MathSciNetCrossRefMATHGoogle Scholar
  11. [DK78a]
    H. Dym and N. Kravitsky, On recovering the mass distribution of a string from its spectral function, Topics in Functional Analysis (Essays Dedicated to M. G. Kreĭn on the Occasion of his 70th Birthday). Vol. 3, Academic Press, New York-London, 1978, pp. 45–90.Google Scholar
  12. [DK78b]
    H. Dym and N. Kravitsky, On the inverse spectral problem for the string equation, Integral Equations Operator Theory 1 (1978), 270–277.MathSciNetCrossRefMATHGoogle Scholar
  13. [DM70]
    H. Dym and H. P. McKean, Application of de Branges spaces of integral functions to the prediction of stationary Gaussian processes, Illinois J. Math. 14 (1970), 299–343.MathSciNetMATHGoogle Scholar
  14. [Fre69]
    G. Freud, Orthogonale Polynome, Birkhäuser Verlag, Basel-Stuttgart, 1969.CrossRefMATHGoogle Scholar
  15. [GL51]
    I. M. Gel'fand and B. M. Levitan, On the determination of a differential equation from its spectral function, Izvestiya Akad. Nauk SSSR. Ser. Mat. 15 (1951), 309–360.MathSciNetMATHGoogle Scholar
  16. [Gol62]
    A. A. Gol'dberg, The integral over a semi-additive measure and its application to the theory of entire functions. II, Mat. Sb. (N.S.) 61 (1962), 334–349.MathSciNetMATHGoogle Scholar
  17. [HM03a]
    V. Havin and J. Mashreghi, Admissible majorants for model subspaces of H2. I. Slow winding of the generating inner function, Canad. J. Math. 55 (2003), 1231–1263.MathSciNetCrossRefMATHGoogle Scholar
  18. [HM03b]
    V. Havin and J. Mashreghi, Admissible majorants for model subspaces of H2. II. Fast winding of the generating inner function, Canad. J. Math. 55 (2003), 1264–1301.MathSciNetCrossRefMATHGoogle Scholar
  19. [KW05a]
    M. Kaltenbäck and H. Woracek, de Branges spaces of exponential type: general theory of growth, Acta Sci. Math. (Szeged) 71 (2005), 231–284.MathSciNetMATHGoogle Scholar
  20. [KW05b]
    M. Kaltenbäck and H. Woracek, Hermite-Biehler functions with zeros close to the imaginary axis, Proc. Amer. Math. Soc. 133 (2005), 245–255 (electronic).MathSciNetCrossRefMATHGoogle Scholar
  21. [LW13a]
    M. Langer and H. Woracek, Indefinite Hamiltonian systems whose Titchmarsh–Weyl coefficients have no finite generalized poles of nonpositive type, Oper. Matrices 7 (2013), 477–555.MathSciNetCrossRefMATHGoogle Scholar
  22. [LW13b]
    M. Langer and H. Woracek, The exponential type of the fundamental solution of an indefinite Hamiltonian system, Complex Anal. Oper. Theory 7 (2013), 285–312.MathSciNetCrossRefMATHGoogle Scholar
  23. [LG86]
    P. Lelong and L. Gruman, Entire Functions of Several Complex Variables, Springer-Verlag, Berlin, 1986.CrossRefMATHGoogle Scholar
  24. [Lev80]
    B. Ja. Levin, Distribution of Zeros of Entire Functions, American Mathematical Society, Providence, RI, 1980.Google Scholar
  25. [OS02]
    J. Ortega-Cerdà and K. Seip, Fourier frames, Ann. of Math. (2) 155 (2002), 789–806.MathSciNetCrossRefMATHGoogle Scholar
  26. [Pit72]
    L. D. Pitt, On problems of trigonometrical approximation from the theory of stationary Gaussian processes, J. Multivariate Anal. 2 (1972), 145–161.MathSciNetCrossRefGoogle Scholar
  27. [Pol13]
    A. Poltoratski, A problem on completeness of exponentials, Ann. of Math. (2) 178 (2013), 983–1016.MathSciNetCrossRefMATHGoogle Scholar
  28. [Rem02]
    C. Remling. Schrödinger operators and de Branges spaces, J. Funct. Anal. 196 (2002), 323–394.MathSciNetCrossRefMATHGoogle Scholar
  29. [Rub96]
    L. A. Rubel, Entire and Meromorphic Functions, Springer-Verlag, New York, 1996.CrossRefMATHGoogle Scholar
  30. [Win00]
    H. Winkler, Small perturbations of canonical systems, Integral Equations Operator Theory 38 (2000), 222–250.MathSciNetCrossRefMATHGoogle Scholar
  31. [Win95]
    H. Winkler, The inverse spectral problem for canonical systems, Integral Equations Operator Theory 22 (1995), 360–374.MathSciNetCrossRefMATHGoogle Scholar
  32. [Wor14]
    H. Woracek, Reproducing kernel almost Pontryagin spaces 40 pp., ASC Report 14. https://doi.org/www.asc.tuwien.ac.at/preprint/2014/asc14x2014.pdf. Vienna University of Technology, 2014.MATHGoogle Scholar
  33. [Yud00]
    P. Yuditskii, Analytic perturbation preserves determinacy of infinite index, Math. Scand. 86 (2000), 288–292.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Hebrew University Magnes Press 2018

Authors and Affiliations

  1. 1.Institute for Analysis and Scientific ComputingVienna University of TechnologyWienAustria

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