Previous |  Up |  Next


boundary behavior of power series; exceptional set
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 $E(f)=E$.
[1] J.  Globevink: Holomorphic functions which are highly nonintegrable at the boundary. Israel J. Math (to appear). MR 1749678
[2] J. Globevnik and E. L. Stout: Highly noncontinuable functions on convex domains. Bull. Sci. Math. 104 (1980), 417–439. MR 0602409
[3] J. Globevnik and E. L. Stout: Holomorphic functions with highly noncontinuable boundary behavior. J. Anal. Math. 41 (1982), 211–216. MR 0687952
[4] J.  Siciak: Highly noncontinuable functions on polynomially convex sets. Zeszyty Naukowe Uniwersytetu Jagiellonskiego 25 (1985), 95–107. MR 0837828 | Zbl 0585.32012
[5] W.  Rudin: Function Theory in the Unit Ball of  $ \mathbb{C}^{n} $. Springer, New York, 1980. MR 0601594
[6] P.  Wojtaszczyk: On highly nonintegrable functions and homogeneous polynomials. Ann. Pol. Math. 65 (1997), 245–251. MR 1441179 | Zbl 0872.32001
Partner of
EuDML logo