An analytic continuation of random analytic functions (in Ukrainian)

Author P. V. Filevych
filevych@mail.ru
Львiвський нацiональний унiверситет ветеринарної медицини та бiотехнологiй iм. С. З. Ґжицького

Abstract Let $(\eta_n(\omega))$ be a sequence of independent random variables such that $\eta_n(\omega)$ takes the values $-1$ and $1$ with the probabilities $p_n$ and $1-p_n$, respectively. Put $q_n=\min\{p_n,1-p_n\}$. Then, for each complex sequence $(a_n)$ such that $\varlimsup\limits_{n\to\infty}\root{n}\of{|a_n|}=1$, the circle $\{z\in\mathbb{C}\colon |z|=1\}$ is the natural boundary for the function $f_\omega(z)=\sum_{n=0}^\infty a_n\eta_n(\omega)z^n$ almost surely if and only if the condition $\sum_{k=0}^\infty q_{n_k}=+\infty$ holds for every increasing sequence $(n_k)$ of nonnegative integers such that $\varliminf\limits_{k\to\infty}\frac{n_k}k<+\infty$.
Keywords random analytic function; singular point; analytic continuation; natural boundary
Reference 1. Fabry E. Sur les points singuliers d'une fonction donnee par son developpement de Taylor// Ann. ec. norm. sup. Paris (3). - 1896. - V.13. - P. 367-399.

 2. Polya G. On converse gap theorems// Trans. Amer. Math. Soc. - 1942. - V.52. - P. 65-71.

 3. Steinhaus H. Uber die Wahrscheinlichkeit dafur, dass der Konvergenzreis einer Potenzreihe ihre naturliche Grence ist// Math. Z. - 1929. - V.31. - P. 408-416.

 4. Paley R.E.A.C., Zygmund A. A note on analytic functions in the unit circle// Proc. Cambridge Phil. Soc. - 1932. - V.28. - P. 266-272.

 5. Kahane J.-P. Some random series of functions. Sec. ed. - Cambridge stud. in adv. math. 5. - Cambridge Univ. Press, 1985. - 308 p.

 6. Ryll-Nardzewski C.D. Blackwell's conjecture on power series with random coefficients// Stud. Math. - 1953. - V.13. - P. 30-36.

 7. Filevych P.V. On the Phragmen-Lindelof Indicator for Random Entire Functions// Ukrainian Mathematical Journal. - 2000. - V.52, №10. - P. 1431-1434. (in Ukrainan)

 8. Filevych P.V. The indicator of entire functions with rapidly oscillating coefficients// Mat. Stud. - 2011. - V.35, №2. - P. 142-148.
Pages 128-132
Volume 36
Issue 2
Year 2011
Journal Matematychni Studii
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