Journal of Mathematical Sciences

, Volume 107, Issue 4, pp 4108–4124

Reproducing Kernels and Extremal Functions in Dirichlet-Type Spaces

  • S. M. Shimorin


Some identities are established for reproducing kernels and extremal functions in Dirichlet-type spaces. In certain cases, these identities are characteristic. A new proof of Aleman's theorem on extremal functions as contractive multipliers is obtained. In contrast to the original one, this proof can be extended to vector-valued spaces. Bibliography: 11 titles.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    A. Aleman, The Multiplication Operator on Hilbert Spaces of Analytic Functions, Habilitationsschrift, Hagen (1993).Google Scholar
  2. 2.
    A. Aleman, S. Richter, and C. Sundberg, “Beurling's theorem for the Bergman space,” Acta Math., 177, 275–310 (1996).Google Scholar
  3. 3.
    P. Duren, D. Khavinson, H. Shapiro, and C. Sundberg, “Contractive zero-divisors in Bergman spaces,” Pacific J.Math., 157, 37–56 (1993).Google Scholar
  4. 4.
    H. Hedenmalm, “A factorization theorem for square area-integrable analytic functions,” J.Reine Angew.Math., 422, 45–68 (1991).Google Scholar
  5. 5.
    N. Nikolskii and V. Vasyunin, “Quasiorthogonal decomposition with respect to complementary metrics and estimates of univalent functions,” St.Petersburg Math.J., 2, (1991).Google Scholar
  6. 6.
    S. Richter, “A representation theorem for cyclic analytic two-isometries,” Trans.Am.Math.Soc., 328, 325–349 (1991).Google Scholar
  7. 7.
    S. Richter, “Invariant subspaces of the Dirichlet shift,” J.Reine Angew.Math., 386, 205–220 (1988).Google Scholar
  8. 8.
    S. Richter and C. Sundberg, “A formula for the local Dirichlet integral,” Michigan Math.J., 38, 355–379 (1991).Google Scholar
  9. 9.
    S. Richter and S. Sundberg, “Invariant subspaces of the Dirichlet shift and pseudocontinuations,” Trans.Am.Math.Soc., 341, 863–879 (1994).Google Scholar
  10. 10.
    S. Shimorin, “Approximative spectral synthesis in the Bergman space,” Preprint, Bordeaux (1997).Google Scholar
  11. 11.
    S. Shimorin, “Factorization of analytic functions in weighted Bergman spaces,” St.Petersburg Math.J., 5, 1005–1022 (1994).Google Scholar

Copyright information

© Plenum Publishing Corporation 2001

Authors and Affiliations

  • S. M. Shimorin
    • 1
  1. 1.St.Petersburg State UniversityRussia

Personalised recommendations