Advertisement

Ukrainian Mathematical Journal

, Volume 58, Issue 4, pp 596–618 | Cite as

Representation of holomorphic functions of many variables by Cauchy-Stieltjes-type integrals

  • V. V. Savchuk
Article
  • 37 Downloads

Abstract

We consider functions of many complex variables that are holomorphic in a polydisk or in the upper half-plane. We give necessary and sufficient conditions under which a holomorphic function is a Cauchy-Stieltjes-type integral of a complex charge. We present several applications of this criterion to integral representations of certain classes of holomorphic functions.

Keywords

Lebesgue Measure Holomorphic Function Borel Measure Weak Topology Multidimensional Case 
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. 1.
    J. A. Shohat and J. D. Tamarkin, The Problem of Moments, Vol. 2, American Mathematical Society, New York (1943).zbMATHGoogle Scholar
  2. 2.
    N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in a Hilbert Space [in Russian], Nauka, Moscow (1966).Google Scholar
  3. 3.
    M. G. Krein and A. A. Nudel’man, Problem of Markov Moments and Extremal Problems [in Russian], Nauka, Moscow (1973).Google Scholar
  4. 4.
    G. M. Goluzin, Geometric Theory of Functions of Complex Variables [in Russian], Nauka, Moscow (1966).Google Scholar
  5. 5.
    L. A. Aizenberg, Carleman Formulas in Complex Analysis [in Russian], Nauka, Novosibirsk (1990).Google Scholar
  6. 6.
    B. Ya. Levin, Lectures on Entire Functions, American Mathematical Society, Providence, RI (1996).zbMATHGoogle Scholar
  7. 7.
    A. Koranyi and J. Pukansky, “Holomorphic functions with positive real part in polycylinder,” Trans. Amer. Math. Soc., 108, 449–456 (1963).zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    V. S. Vladimirov and Yu. V. Drozhzhinov, “Holomorphic functions in a polydisk with nonnegative imaginary part,” Mat. Zametki, 15, No. 1, 55–61 (1974).zbMATHMathSciNetGoogle Scholar
  9. 9.
    L. A. Aizenberg and Sh. A. Dautov, “Holomorphic functions of many complex variables with nonnegative real part. Traces of holomorphic and pluriharmonic functions on the Shilov boundary,” Mat. Sb., 99, No. 3, 342–355 (1976).MathSciNetGoogle Scholar
  10. 10.
    S. Kosbergenov and A. M. Kytmanov, “Generalization of the Schwartz and Riesz-Herglotz formulas in Reinhart domains,” Izv. Vyssh. Uchebn. Zaved., Ser. Mat., No. 10, 60–63 (1984).Google Scholar
  11. 11.
    V. S. Vladimirov and A. G. Sergeev, “Complex analysis in a pipe of future,” in: VINITI Series in Contemporary Problems of Mathematics (Fundamental Trends) [in Russian], Vol. 8, VINITI, Moscow (1985), pp. 191–266.Google Scholar
  12. 12.
    G. Ts. Tumarkin, “On Cauchy-Stieltjes-type integrals,” Usp. Mat. Nauk, 11, No. 4, 163–166 (1956).zbMATHMathSciNetGoogle Scholar
  13. 13.
    P. Duren, Theory of Hp Spaces, Academic Press, New York (1970).Google Scholar
  14. 14.
    W. K. Hayman and P. B. Kennedy, Subharmonic Functions [Russian translation], Mir, Moscow (1980).zbMATHGoogle Scholar
  15. 15.
    L. A. Lyusternik and V. I. Sobolev, Elements of Functional Analysis [in Russian], Nauka, Moscow (1965).zbMATHGoogle Scholar
  16. 16.
    W. Rudin, Function Theory in Polydiscs [Russian translation], Mir, Moscow (1974).zbMATHGoogle Scholar
  17. 17.
    E. Hille and J. D. Tamarkin, “On absolute integrability of Fourier transforms,” Fund. Math., 25, 329–352 (1935).zbMATHGoogle Scholar
  18. 18.
    M. A. Evgrafov, Analytic Functions [in Russian], Nauka, Moscow (1991).zbMATHGoogle Scholar
  19. 19.
    L. N. Znamenskaya, “Generalization of the F. and M. Riesz theorem and Carleman formula,” Sib. Mat. Zh., 29, No. 4, 75–79 (1988).zbMATHMathSciNetGoogle Scholar
  20. 20.
    L. N. Znamenskaya, “Multidimensional analogs of the F. and M. Riesz theorem and Carleman formula, ” Izv. Vyssh. Uchebn. Zaved., Ser. Mat., No. 7, 67–69 (1989).Google Scholar
  21. 21.
    M. M. Roginskaya, “Two multidimensional analogs of the F. and M. Riesz theorem,” Zap. Nauchn. Sem. Peterburg. Otdel. Mat. Inst. Ros. Akad. Nauk, 255, 16–176 (1998).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • V. V. Savchuk
    • 1
  1. 1.Institute of MathematicsUkrainian Academy of SciencesKyiv

Personalised recommendations