Zeros of random functions in Bergman spaces

Shapiro, Joel H.. "Zeros of random functions in Bergman spaces." Annales de l'institut Fourier 29.4 (1979): 159-171. <http://eudml.org/doc/74429>.

@article{Shapiro1979,
abstract = {Suppose $\mu $ is a finite positive rotation invariant Borel measure on the open unit disc $\Delta $, and that the unit circle lies in the closed support of $\mu $. For $0&lt; p&lt; \infty $ the Bergman space$A^p_\mu $ is the collection of functions in $L^p(\mu )$ holomorphic on $\Delta $. We show that whenever a Gaussian power series $f(z) = \Sigma \zeta _na_nz^n$ almost surely lies in $A^p_\mu $ but not in $\bigcup _\{q&gt;p\} A^p_\mu $, then almost surely: a) the zero set $Z(f)$ of $f$ is not contained in any $A^q_\mu $ zero set ($q&gt;p$, and b) $Z(f+1)\cup Z(f-1)$ is not contained in any $A^q_\mu $ zero set.},
author = {Shapiro, Joel H.},
journal = {Annales de l'institut Fourier},
keywords = {Random Functions; Bergman Spaces; Rotation Invariant Borel Measure; Gaussian Power Series},
language = {eng},
number = {4},
pages = {159-171},
publisher = {Association des Annales de l'Institut Fourier},
title = {Zeros of random functions in Bergman spaces},
url = {http://eudml.org/doc/74429},
volume = {29},
year = {1979},
}

TY - JOUR
AU - Shapiro, Joel H.
TI - Zeros of random functions in Bergman spaces
JO - Annales de l'institut Fourier
PY - 1979
PB - Association des Annales de l'Institut Fourier
VL - 29
IS - 4
SP - 159
EP - 171
AB - Suppose $\mu $ is a finite positive rotation invariant Borel measure on the open unit disc $\Delta $, and that the unit circle lies in the closed support of $\mu $. For $0&lt; p&lt; \infty $ the Bergman space$A^p_\mu $ is the collection of functions in $L^p(\mu )$ holomorphic on $\Delta $. We show that whenever a Gaussian power series $f(z) = \Sigma \zeta _na_nz^n$ almost surely lies in $A^p_\mu $ but not in $\bigcup _{q&gt;p} A^p_\mu $, then almost surely: a) the zero set $Z(f)$ of $f$ is not contained in any $A^q_\mu $ zero set ($q&gt;p$, and b) $Z(f+1)\cup Z(f-1)$ is not contained in any $A^q_\mu $ zero set.
LA - eng
KW - Random Functions; Bergman Spaces; Rotation Invariant Borel Measure; Gaussian Power Series
UR - http://eudml.org/doc/74429
ER -