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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Boundary zero sets of $ A^\infty$ functions satisfying growth conditions


Authors: B. A. Taylor and D. L. Williams
Journal: Proc. Amer. Math. Soc. 35 (1972), 155-160
MSC: Primary 30A72
DOI: https://doi.org/10.1090/S0002-9939-1972-0310253-9
MathSciNet review: 0310253
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Abstract: Let A denote the algebra of functions analytic in the open unit disc D and continuous in D, and let

$\displaystyle {A^\infty } = \{ f \in A:{f^{(n)}} \in A,n = 0,1,2, \cdots \} .$

For $ f \in A$ denote the set of zeros of f in D by $ {Z^0}(f)$, and for $ f \in {A^\infty }$ let $ {Z^\infty }(f) = \bigcap {_{n = 0}^\infty Z{^0}({f^{(n)}})} $. We study the boundary zero sets of $ {A^\infty }$ functions F satisfying, for some sequence $ \{ {M_n}\} $ and some $ B > 0$,

$\displaystyle \vert{F^{(n)}}(z)\vert \leqq n!{B^n}{M_n},\quad z \in \bar D,n = 0,1,2, \cdots .$ ($ 1$)

In particular, when $ {M_n} = \exp ({n^p}),p > 1$, it is shown that for E, a proper closed subset of $ \partial D$, there exists $ F \in {A^\infty }$ satisfying (1) and with $ {Z^0}(F) = {Z^\infty }(F) = E$ if and only if $ \smallint _{ - \pi }^\pi \vert\log \rho ({e^{i\theta }},E){\vert^q}D\theta < + \infty $. Here $ \rho (z,E)$ is the distance from z to E and $ (1/p) + (1/q) = 1$.

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DOI: https://doi.org/10.1090/S0002-9939-1972-0310253-9
Keywords: Boundary zeros of analytic functions, sets of uniqueness, spaces of analytic functions, quasi-analytic classes
Article copyright: © Copyright 1972 American Mathematical Society