On entire functions of exponential type and indicators of analytic functionals

C. O. Kiselman The University of Stockholm, Stockholm, Sweden

TBD mathscidoc:1701.331310

Acta Mathematica, 117, (1), 1-35, 1966.5
We shall be concerned with the indicator$p$of an analytic functional μ on a complex manifold$U$: $$p(\varphi ) = \overline {\mathop {\lim }\limits_{t \to + \infty } } \frac{l}{t}\log \left| {\mu (e^{t\varphi } )} \right|,$$ where ϕ is an arbitrary analytic function on$U$. More specifically, we shall consider the smallest upper semicontinuous majorant$p$^{$J$}of the restriction of$p$to a subspace £ of the analytic functions. An obvious problem is then to characterize the set of functions$p$^{$J$}which can occur as regularizations of indicators. In the case when$U$=$C$^{$n$}and £ is the space of all linear functions on$C$^{$n$}, this set can be described more easily as the set of functions(0.1) $$\mathop {\lim }\limits_{\theta \to \zeta } \overline {\mathop {\lim }\limits_{t \to + \infty } } \frac{l}{t}\log \left| {u(t\theta )} \right|$$ of$n$complex variables ζ∈$C$^{$n$}where$u$is an entire function of exponential type in$C$^{$n$}. We hall prove that a function in$C$^{$n$}is of the form (0.1) for some entire function$u$of exponential type if and only if it is plurisubharmonic and positively homogeneous of order one (Theorem 3.4). The proof is based on the characterization given by Fujita and Takeuchi of those open subsets of complex projective$n$-space which are Stein manifolds.
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@inproceedings{c.1966on,
  title={On entire functions of exponential type and indicators of analytic functionals},
  author={C. O. Kiselman},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203209725156019},
  booktitle={Acta Mathematica},
  volume={117},
  number={1},
  pages={1-35},
  year={1966},
}
C. O. Kiselman. On entire functions of exponential type and indicators of analytic functionals. 1966. Vol. 117. In Acta Mathematica. pp.1-35. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203209725156019.
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