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The Bergman Kernel and a Generalized Fourier-Borel Transform

  • Friedrich Haslinger
Part of the International Society for Analysis, Applications and Computation book series (ISAA, volume 3)

Abstract

In this paper we represent the dual space of a Fréchet space of entire functions again as a space of entire functions. For this purpose we use the Bergman kernel of a certain Hilbert space. In the classical setting the exponential functions provide the isomorphism via Fourier-Borel transform. In our case we use the Bergman kernel instead of the exponential functions in order to establish the isomorphism.

Keywords

Hilbert Space Exponential Function Entire Function Dual Space Toeplitz Operator 
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.

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References

  1. [BD]
    C.A. Berenstein and M.A. Dostal Analytically uniform spaces and their applications to convolution equations Lecture Notes in Mathematics, Vol. 256, Springer Verlag, Berlin, 1972.Google Scholar
  2. [BT]
    C.A. Berenstein and B.A. Taylor A new look at interpolation theory for entire functions of one variable Adv. in Math. 33 1979, pp. 109–143.CrossRefMathSciNetzbMATHGoogle Scholar
  3. [Bo]
    R.P. Boas Entire functions Academic Press, New York, 1954.Google Scholar
  4. [E]
    L. Ehrenpreis Fourier analysis in several complex variables Wiley-Interscience, New York, 1970.Google Scholar
  5. [FH]
    G. Francsics and N. Hanges Explicit formulas for the Szegô kernel on certain weakly pseudoconvex domains Proc.Amer.Math.Soc. 304 1995, pp. 3161–3168.MathSciNetGoogle Scholar
  6. [GS]
    P. C. Greiner and E. M. Stein On the solvability of some differential operators of type Ob Proc. Internat. Conf., (Cortona, Italy, 1976–1977), Scuola Norm. Sup. Pisa, 1978, pp. 106–165.Google Scholar
  7. [H1]
    F. Haslinger Weighted spaces of entire functions Indiana Univ. Math. J. 35 1986, pp. 193–207.Google Scholar
  8. [H2]
    F. Haslinger Szegô kernels of certain unbounded domains in C2 Révue Bourn. Math. Pures et Appl. 39 1994, pp. 914–926.MathSciNetGoogle Scholar
  9. [H3]
    F. Haslinger Singularities of the Szegô kernel for certain weakly pseudoconvex domains in C 2 J. Functional Analysis 129 1995, pp. 406–427.CrossRefMathSciNetzbMATHGoogle Scholar
  10. [H4]
    F. Haslinger Bergman and Hardy spaces on model domains Illinois J. of Math. (to appear)Google Scholar
  11. [K]
    S.G. Krantz Function theory of several complex variables Wadsworth Brooks/Cole, Pacific Grove, CA, 1992.Google Scholar
  12. [LG]
    P. Lelong and L. Gruman Entire functions of several complex variables Grundlehren der mathematischen Wissenschaften Vol. 282, Springer-Verlag, Berlin-New York, 1986.Google Scholar
  13. [Ml]
    A. Martineau Sur les fonctionelles analytiques et la transformation de Fourier-Borel J.d’Analyse Math. 9 (1963), pp. 1–144.MathSciNetGoogle Scholar
  14. [M2]
    A. Martineau Equations différentielles d’ordre infini Bull. Soc. math. France 95 (1967), pp. 109–154.MathSciNetzbMATHGoogle Scholar
  15. [P]
    A. Pietsch Nukleare lolealkonvexe Raume Akademie-Verlag, Berlin, 1965.Google Scholar
  16. [R]
    M. Range Holomorphic functions and integral representations in several complex variables Springer-Verlag, Berlin, 1986.Google Scholar
  17. [S]
    S. Saitoh Integral transforms, reproducing kernels and their applications Addison, Pitman Research Notes in Mathematics, Vol. 369, 1997.Google Scholar
  18. [T]
    B.A. Taylor On weighted polynomial approximation of entire functions Pacific J. Math. 36 (1971), pp. 523–536.Google Scholar
  19. [W]
    J. Wloka Reproduzierende Kerne und nukleare Raume Math. Ann. 163 (1966), pp. 167–188.CrossRefMathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1999

Authors and Affiliations

  • Friedrich Haslinger
    • 1
  1. 1.Institut für MathematikUniversität WienAustria

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