Generalization of weierstrass canonical integrals

Olga Veselovska

Open Mathematics (2004)

  • Volume: 2, Issue: 4, page 593-604
  • ISSN: 2391-5455

Abstract

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In this paper we prove that a subharmonic function in ℝm of finite λ-type can be represented (within some subharmonic function) as the sum of a generalized Weierstrass canonical integral and a function of finite λ-type which tends to zero uniformly on compacts of ℝm. The known Brelot-Hadamard representation of subharmonic functions in ℝm of finite order can be obtained as a corollary from this result. Moreover, some properties of R-remainders of λ-admissible mass distributions are investigated.

How to cite

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Olga Veselovska. "Generalization of weierstrass canonical integrals." Open Mathematics 2.4 (2004): 593-604. <http://eudml.org/doc/268723>.

@article{OlgaVeselovska2004,
abstract = {In this paper we prove that a subharmonic function in ℝm of finite λ-type can be represented (within some subharmonic function) as the sum of a generalized Weierstrass canonical integral and a function of finite λ-type which tends to zero uniformly on compacts of ℝm. The known Brelot-Hadamard representation of subharmonic functions in ℝm of finite order can be obtained as a corollary from this result. Moreover, some properties of R-remainders of λ-admissible mass distributions are investigated.},
author = {Olga Veselovska},
journal = {Open Mathematics},
keywords = {31B05},
language = {eng},
number = {4},
pages = {593-604},
title = {Generalization of weierstrass canonical integrals},
url = {http://eudml.org/doc/268723},
volume = {2},
year = {2004},
}

TY - JOUR
AU - Olga Veselovska
TI - Generalization of weierstrass canonical integrals
JO - Open Mathematics
PY - 2004
VL - 2
IS - 4
SP - 593
EP - 604
AB - In this paper we prove that a subharmonic function in ℝm of finite λ-type can be represented (within some subharmonic function) as the sum of a generalized Weierstrass canonical integral and a function of finite λ-type which tends to zero uniformly on compacts of ℝm. The known Brelot-Hadamard representation of subharmonic functions in ℝm of finite order can be obtained as a corollary from this result. Moreover, some properties of R-remainders of λ-admissible mass distributions are investigated.
LA - eng
KW - 31B05
UR - http://eudml.org/doc/268723
ER -

References

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  1. [1] L.I. Ronkin: Introduction into the theory of entire functions of several variables, Nauka, Moscow, 1971. [Russian] 
  2. [2] A. Bateman, A. Erdelyi: Higher transcendental functions, 2, Nauka, 1974. [Russian] Zbl0143.29202
  3. [3] W.K. Hayman, R.B. Kennedy: Subharmonic functions, Acad. Press, London, 1976. Zbl0419.31001
  4. [4] A.A. Kondratyuk: “Spherical harmonics and subharmonic functions (Russian)”, Mat. Sb., Vol. 125, (1984), pp. 147–166. [English translation in Math. USSR, Sb. 53, (1986), pp. 147–167] 
  5. [5] B. Azarin: Theory of growth of subharmonic functions, Texts of lecture, part 2, Krakov, (1982). [Russian] 
  6. [6] vL.A. Rubel: A generalized canonical product In trans: Modern problems of theory of analytic functions, Nauka, Moscow (1966), pp. 264–270. 
  7. [7] Ya. V. Vasylkiv, An investigation asymptotic characteristics of entire and subharmonic functions by method of Fourier series, Abstract dissertation, Donetsk, 1986. (Russian) 
  8. [8] E. Stein, G. Weiss: Introduction to Fourier analysis of Euclidean spaces, Princeton University Press, Princeton, New Jersey, 1971. Zbl0232.42007
  9. [9] H. Berens, P.L. Butzer, S. Pawelke: “Limitierungs verfahren von Reihen mehrdimensionaler Kugelfunktionen und deren Saturationsverhalten”, Publs. Res. Inst. Math. Sci., Vol. 4, (1968), pp. 201–268. Zbl0201.08402
  10. [10] A.A. Kondratyuk: “On the method of spherical harmonics for subharmonic functions (Russian)”, Mat. Sb., Vol. 116, (1981), pp. 147–165. [English translation in Math. USSR, Sb. 44, (1983), pp. 133–148] Zbl0473.31007
  11. [11] O.V. Veselovska: “Analog of Miles theorem for δ-subharmonic functions in ℝm ”, Ukr. Math. J., Vol. 36, (1984), pp. 694–698. [Ukrainian] 
  12. [12] N.N. Lebedev: Special functions and their applications, Revised edition, translated from the Russian and edited by Richard A. Silverman, Dover Publications, Inc., New York, 1972. 
  13. [13] L.I. Ronkin: Functions of completely regular growth, translated from the Russian by A. Ronkin and I. Yedvabnik, Mathematics and its Applications (Soviet Series), Vol. 81, Kluwer Academic Publishers Group, Dordrecht, 1992. 

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