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Asymptotically holomorphic functions and certain of their applications. (English) Zbl 0744.30038
Proc. Int. Congr. Math., Kyoto/Japan 1990, Vol. II, 959-967 (1991).
[For the entire collection see Zbl 0741.00020.]
The classical Phragmén-Lindelöf theorems are of the form: If $$f(z)\in H(D)$$ satisfies certain conditions on the boundary $$\partial D$$ of $$D$$, then either $$f(z)$$ is bounded on $$D$$ or $$f(z)$$ must be rather big somewhere in $$D$$. The author’s theory of “asymptotically holomorphic” $$(AH)$$ functions gives analogues of these theorems for the much wider class of $$AH$$-functions.
An $$AH$$-function $$f(z)$$ defined in $$D$$ is a $$C^ 1$$-function subject to two types of conditions: 1) growth conditions on $$f$$ and 2) a condition enforcing that $$\overline\partial f$$ tends to zero at a prescribed (rather fast) rate as $$z\to\partial D$$. Three precise theorems are stated for the cases of $$D$$=disk, $$D$$=half-plane, $$D$$=punctured disk.
Three important applications of these theorems are explained by the author. They are 1. Far-reaching generalizations of Beurling’s well-known characterization of the shift-invariant subspaces of the Hardy space $$H^ 2(U)$$.
2. The determination of the closure of $$\mathbb{C}[z]$$ in the Banach space $$L^ p(\mu)$$ $$(1\leq p<\infty)$$ with the norm $$\| g\|^ p=\int| g|^ pd\mu$$, where $$\mu$$ is a positive Borel measure with compact support [J. E. Thomson, Ann. Math., II. Ser. 133, 477- 507 (1991; Zbl 0736.41008)]. 3. Proof that the differential equation $$\dot x=\alpha(x,y)$$, $$\dot y=\beta(x,y)$$ ($$\alpha$$, $$\beta$$ quasi- analytic functions subject to growth conditions) has only a finite number of limit cycles in any compact region of the plane, provided that all singular points are non-degenerate [A. L. Volberg, Publ. Math. Orsay, Sémin. Anal. Harmonique 1989/90, 152-171 (1990; Zbl 0719.34052)].

##### MSC:
 30G30 Other generalizations of analytic functions (including abstract-valued functions)
##### Keywords:
Phragmén-Lindelöf theorems; Hardy space; limit cycles