TY - JOUR

T1 - TWO PROBLEMS ON RANDOM ANALYTIC FUNCTIONS IN FOCK SPACES

AU - Fang, Xiang

AU - Tien, Pham Trong

N1 - Publisher Copyright:
© 2022 Cambridge University Press. All rights reserved.

PY - 2022

Y1 - 2022

N2 - Let f(z) = Σ∞n=0anznbe an entire function on the complex plane and let Rf(z) = Σ∞n=0anXnznbe its randomization induced by a standard sequence (Xn)nof independent Bernoulli, Steinhaus or Gaussian random variables. In this paper, we characterize those functions f(z) such that Rf(z) is almost surely in the Fock space Fpαfor any p; α ∈ (0, ∞). Then such a characterization, together with embedding theorems which are of independent interests, is used to obtain a Littlewood-type theorem, aka regularity improvement under randomization within the scale of Fock spaces. Other results obtained in this paper include (a) a characterization of random analytic functions in the mixed norm space F(∞; q; α), an endpoint version of Fock spaces, via entropy integrals; (b) a complete description of random lacunary elements in Fock spaces; (c) a complete description of random multipliers between different Fock spaces.

AB - Let f(z) = Σ∞n=0anznbe an entire function on the complex plane and let Rf(z) = Σ∞n=0anXnznbe its randomization induced by a standard sequence (Xn)nof independent Bernoulli, Steinhaus or Gaussian random variables. In this paper, we characterize those functions f(z) such that Rf(z) is almost surely in the Fock space Fpαfor any p; α ∈ (0, ∞). Then such a characterization, together with embedding theorems which are of independent interests, is used to obtain a Littlewood-type theorem, aka regularity improvement under randomization within the scale of Fock spaces. Other results obtained in this paper include (a) a characterization of random analytic functions in the mixed norm space F(∞; q; α), an endpoint version of Fock spaces, via entropy integrals; (b) a complete description of random lacunary elements in Fock spaces; (c) a complete description of random multipliers between different Fock spaces.

KW - Fock spaces

KW - Random analytic functions

UR - http://www.scopus.com/inward/record.url?scp=85134051757&partnerID=8YFLogxK

U2 - 10.4153/S0008414X22000372

DO - 10.4153/S0008414X22000372

M3 - 期刊論文

AN - SCOPUS:85134051757

JO - Canadian Journal of Mathematics

JF - Canadian Journal of Mathematics

SN - 0008-414X

ER -