Description of simple exceptional sets in the unit ball

Kot, Piotr. "Description of simple exceptional sets in the unit ball." Czechoslovak Mathematical Journal 54.1 (2004): 55-63. <http://eudml.org/doc/30836>.

@article{Kot2004,
abstract = {For $ z\in \partial B^n$, the boundary of the unit ball in $\mathbb \{C\}^n$, let $\Lambda (z)=\lbrace \lambda \:|\lambda |\le 1\rbrace $. If $ f\in \mathbb \{O\}(B^n)$ then we call $E(f)=\lbrace z\in \partial B^n\:\int _\{\Lambda (z)\}|f(z)|^2\mathrm \{d\}\Lambda (z)=\infty \rbrace $ the exceptional set for $f$. In this note we give a tool for describing such sets. Moreover we prove that if $E$ is a $G_\delta $ and $F_\sigma $ subset of the projective $(n-1)$-dimensional space $\mathbb \{P\}^\{n-1\}=\mathbb \{P\}(\mathbb \{C\}^n)$ then there exists a holomorphic function $f$ in the unit ball $B^n$ so that $E(f)=E$.},
author = {Kot, Piotr},
journal = {Czechoslovak Mathematical Journal},
keywords = {boundary behavior of power series; exceptional set; boundary behavior of power series; exceptional set},
language = {eng},
number = {1},
pages = {55-63},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Description of simple exceptional sets in the unit ball},
url = {http://eudml.org/doc/30836},
volume = {54},
year = {2004},
}

TY - JOUR
AU - Kot, Piotr
TI - Description of simple exceptional sets in the unit ball
JO - Czechoslovak Mathematical Journal
PY - 2004
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 54
IS - 1
SP - 55
EP - 63
AB - For $ z\in \partial B^n$, the boundary of the unit ball in $\mathbb {C}^n$, let $\Lambda (z)=\lbrace \lambda \:|\lambda |\le 1\rbrace $. If $ f\in \mathbb {O}(B^n)$ then we call $E(f)=\lbrace z\in \partial B^n\:\int _{\Lambda (z)}|f(z)|^2\mathrm {d}\Lambda (z)=\infty \rbrace $ the exceptional set for $f$. In this note we give a tool for describing such sets. Moreover we prove that if $E$ is a $G_\delta $ and $F_\sigma $ subset of the projective $(n-1)$-dimensional space $\mathbb {P}^{n-1}=\mathbb {P}(\mathbb {C}^n)$ then there exists a holomorphic function $f$ in the unit ball $B^n$ so that $E(f)=E$.
LA - eng
KW - boundary behavior of power series; exceptional set; boundary behavior of power series; exceptional set
UR - http://eudml.org/doc/30836
ER -