Skip to main content
Log in

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

  • Published:
Ukrainian Mathematical Journal Aims and scope

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. J. A. Shohat and J. D. Tamarkin, The Problem of Moments, Vol. 2, American Mathematical Society, New York (1943).

    MATH  Google Scholar 

  2. N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in a Hilbert Space [in Russian], Nauka, Moscow (1966).

    Google Scholar 

  3. M. G. Krein and A. A. Nudel’man, Problem of Markov Moments and Extremal Problems [in Russian], Nauka, Moscow (1973).

    Google Scholar 

  4. G. M. Goluzin, Geometric Theory of Functions of Complex Variables [in Russian], Nauka, Moscow (1966).

    Google Scholar 

  5. L. A. Aizenberg, Carleman Formulas in Complex Analysis [in Russian], Nauka, Novosibirsk (1990).

    Google Scholar 

  6. B. Ya. Levin, Lectures on Entire Functions, American Mathematical Society, Providence, RI (1996).

    MATH  Google Scholar 

  7. A. Koranyi and J. Pukansky, “Holomorphic functions with positive real part in polycylinder,” Trans. Amer. Math. Soc., 108, 449–456 (1963).

    Article  MATH  MathSciNet  Google Scholar 

  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).

    MATH  MathSciNet  Google Scholar 

  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).

    MathSciNet  Google Scholar 

  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).

  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. G. Ts. Tumarkin, “On Cauchy-Stieltjes-type integrals,” Usp. Mat. Nauk, 11, No. 4, 163–166 (1956).

    MATH  MathSciNet  Google Scholar 

  13. P. Duren, Theory of Hp Spaces, Academic Press, New York (1970).

    Google Scholar 

  14. W. K. Hayman and P. B. Kennedy, Subharmonic Functions [Russian translation], Mir, Moscow (1980).

    MATH  Google Scholar 

  15. L. A. Lyusternik and V. I. Sobolev, Elements of Functional Analysis [in Russian], Nauka, Moscow (1965).

    MATH  Google Scholar 

  16. W. Rudin, Function Theory in Polydiscs [Russian translation], Mir, Moscow (1974).

    MATH  Google Scholar 

  17. E. Hille and J. D. Tamarkin, “On absolute integrability of Fourier transforms,” Fund. Math., 25, 329–352 (1935).

    MATH  Google Scholar 

  18. M. A. Evgrafov, Analytic Functions [in Russian], Nauka, Moscow (1991).

    MATH  Google Scholar 

  19. L. N. Znamenskaya, “Generalization of the F. and M. Riesz theorem and Carleman formula,” Sib. Mat. Zh., 29, No. 4, 75–79 (1988).

    MATH  MathSciNet  Google Scholar 

  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).

  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 

Download references

Author information

Authors and Affiliations

Authors

Additional information

__________

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 4, pp. 522–542, April, 2006.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Savchuk, V.V. Representation of holomorphic functions of many variables by Cauchy-Stieltjes-type integrals. Ukr Math J 58, 596–618 (2006). https://doi.org/10.1007/s11253-006-0086-5

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-006-0086-5

Keywords

Navigation