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Random analytic functions with a prescribed growth rate in the unit disk

Part of: Geometric function theory Spaces and algebras of analytic functions Series expansions

Published online by Cambridge University Press:  26 April 2024

Xiang Fang Open the ORCID record for Xiang Fang [Opens in a new window]  and
Pham Trong Tien Open the ORCID record for Pham Trong Tien [Opens in a new window]
Xiang Fang*
Affiliation:
Department of Mathematics, National Central University, Chungli, Taiwan (R.O.C.)
Pham Trong Tien
Affiliation:
Faculty of Mathematics, Mechanics and Informatics, VNU University of Science, Vietnam National University, Hanoi, Vietnam e-mail: phamtien@vnu.edu.vn
*
e-mail: xfang@math.ncu.edu.tw
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Abstract

Let $\mathcal {R}f$ be the randomization of an analytic function over the unit disk in the complex plane

$$ \begin{align*}\mathcal{R} f(z) =\sum_{n=0}^\infty a_n X_n z^n \in H({\mathbb D}), \end{align*} $$
where $f(z)=\sum _{n=0}^\infty a_n z^n \in H({\mathbb D})$ and $(X_n)_{n \geq 0}$ is a standard sequence of independent Bernoulli, Steinhaus, or complex Gaussian random variables. In this paper, we demonstrate that prescribing a polynomial growth rate for random analytic functions over the unit disk leads to rather satisfactory characterizations of those $f \in H({\mathbb D})$ such that ${\mathcal R} f$ admits a given rate almost surely. In particular, we show that the growth rate of the random functions, the growth rate of their Taylor coefficients, and the asymptotic distribution of their zero sets can mutually, completely determine each other. Although the problem is purely complex analytic, the key strategy in the proofs is to introduce a class of auxiliary Banach spaces, which facilitate quantitative estimates.

Keywords

Random analytic functionsgrowth ratezero setcounting functionBlaschke condition

MSC classification

Primary: 30B20: Random power series
Secondary: 30H99: None of the above, but in this section 30C15: Zeros of polynomials, rational functions, and other analytic functions (e.g. zeros of functions with bounded Dirichlet integral)
Type
Article
Information
Canadian Journal of Mathematics , First View , pp. 1 - 24
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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Footnotes

X. Fang is supported by NSTC of Taiwan (Grant No. 112-2115-M-008-010-MY2).

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