Taiwanese Journal of Mathematics

ON DOMINATING PHENOMENON FOR BOUNDED REAL HARMONIC FUNCTIONS IN SEVERAL VARIABLES

So-Chin Chen and Cin-Chang Lu

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Abstract

In this article, motivated by the work of Brown, Shields, and Zeller[2], we first consider the representation problem of zero by exponential sums in several complex variables. Then we give a complete characterization of dominating sets for bounded real harmonic functions, denoted by $h^\infty(\Re{B_n})$, on the open unit ball $\Re{B_n}$ in $\mathbb{R}^n$, $n \geq 3$, and for bounded real $\mathcal{M}$-harmonic functions, denoted by $\mathcal{M} h^\infty(B_n)$, on the open unit ball $B_n$ in $\mathbb{C}^n$, $n \ge 2$.

Article information

Source
Taiwanese J. Math., Volume 14, Number 2 (2010), 501-515.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500405804

Digital Object Identifier
doi:10.11650/twjm/1500405804

Mathematical Reviews number (MathSciNet)
MR2655784

Zentralblatt MATH identifier
1202.31004

Subjects
Primary: 32A30: Other generalizations of function theory of one complex variable (should also be assigned at least one classification number from Section 30) {For functions of several hypercomplex variables, see 30G35} 30D10: Representations of entire functions by series and integrals

Keywords
dominating sets harmonic functions $\mathcal{M}$-harmonic functions

Citation

Chen, So-Chin; Lu, Cin-Chang. ON DOMINATING PHENOMENON FOR BOUNDED REAL HARMONIC FUNCTIONS IN SEVERAL VARIABLES. Taiwanese J. Math. 14 (2010), no. 2, 501--515. doi:10.11650/twjm/1500405804. https://projecteuclid.org/euclid.twjm/1500405804


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References

  • [1.] F. F. Bonsall, Domination of the supremum of a bounded harmonic function by its supremum over a countable set, Proc. Edinburgh Math. Soc., 30 (1987), 471-477.
  • [2.] L. Brown, A. Shields and K. Zeller, On absolutely convergent exponential sums, Trans. Amer. Math. Soc., 96 (1960), 162-183.
  • [3.] S. C. Chen, On dominating sets for uniform algebra on pseudoconvex domains, Journal of Pure and Applied Mathematics Quarterly, Special issue in honor of J. J. Kohn, in press.
  • [4.] N. Danikas and W. K. Hayman, Domination on sets and in $H^p$, Results Math., 34 (1998), 85-90.
  • [5.] W. K. Hayman, Domination on sets and in Norm, Contemporary Mathematics, 404 (2006), 103-109.
  • [6.] E. Hille and R. S. Phillips, Functional analysis and semi-groups, Amer. Math. Soc. Colloquium Publications, 1957, p. 31.
  • [7.] X. Massaneda and P. J. Thomas, Sampling sequences for Hardy spaces of the ball Proc. Amer. Math. Soc., 128(3) (2000), 837-843.
  • [8.] R. Narasimhan, Several Complex Variables, University of Chicago Press, Chicago, 1971.
  • [9.] W. Rudin, Function Theory in the Unit Ball of ${\Bbb C}^n$, Springer-Verlag, New York, 1980.