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Growth properties of Paley-Wiener functions on $${\mathbb{C}}^ n$$. (English) Zbl 0571.32001
This article is concerned with the growth properties of an entire function f of exponential type over $${\mathbb{C}}^ n$$ which is in $$L^ 2$$ on the real subspace $${\mathbb{R}}^ n$$. It is known that such f is a Paley- Wiener function so that it is the complex Fourier transform of a function g of $$L^ 2({\mathbb{R}}^ n)$$ with compact support $$K\subset {\mathbb{R}}^ n$$. The paper brings to focus various such interesting and relevant results, establishes certain results, some of which also have intrinsic significance and ultimately achieves its set goal to show that f is of ”regular growth” if the convex hull of K is a polyhedron. The ”way clearing” efforts include analysis of (i) the indicators of f; (ii) useful equivalents of the notion of ”regular growth”; and (iii) a ”lower- bounds theorem” concerning a non-positive sub-harmonic function of 3 or more real variables (based on potential theoretic considerations). The lucid style of the article offers reasonable attractions for one to swim through its depths (if one is not mislead by its modesty).
Reviewer: J.Gopala Krishna

##### MSC:
 32A15 Entire functions of several complex variables 32A22 Nevanlinna theory; growth estimates; other inequalities of several complex variables 31B10 Integral representations, integral operators, integral equations methods in higher dimensions