TY - JOUR

T1 - RANDOM ANALYTIC FUNCTIONS WITH A PRESCRIBED GROWTH RATE IN THE UNIT DISK

AU - Fang, Xiang

AU - Tien, Pham Trong

N1 - Publisher Copyright:
© 2024 Cambridge University Press. All rights reserved.

PY - 2024

Y1 - 2024

N2 - Let Rf be the randomization of an analytic function over the unit disk in the complex plane:(Formula presented) Where (Formula presented) and (Formula presented) 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 ϵ H(D) such that Rf 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.

AB - Let Rf be the randomization of an analytic function over the unit disk in the complex plane:(Formula presented) Where (Formula presented) and (Formula presented) 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 ϵ H(D) such that Rf 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.

KW - Blaschke condition

KW - counting function

KW - growth rate

KW - Littlewood-type theorem

KW - Random analytic functions

KW - zero set

UR - http://www.scopus.com/inward/record.url?scp=85191610175&partnerID=8YFLogxK

U2 - 10.4153/S0008414X24000403

DO - 10.4153/S0008414X24000403

M3 - 期刊論文

AN - SCOPUS:85191610175

SN - 0008-414X

JO - Canadian Journal of Mathematics

JF - Canadian Journal of Mathematics

ER -