On separated Carleson sequences in the ball of ${\mathbb{C}}^{n}.$

Details On separated Carleson sequences in the ball of ${\mathbb{C}}^{n}.$ On separated Carleson sequences in the ball of ${\mathbb{C}}^{n}.$

2011-09-19

arxiv.org/abs/1109.4040v1

Mathematics

Complex Variables

Scientific paper

A. Hartmann proved recently that just one bounded holomorphic function in the unit disc ${\mathbb{D}}$ of the complex plane ${\mathbb{C}},$ was enough to characterize interpolating sequences for $H^{\infty}({\mathbb{D}}).$ In this work we generalize this to the case of the unit ball in ${\mathbb{C}}^{n}.$ Precisely to a sequence $S$ of points in ${\mathbb{B}}$ we associate canonically a pair $S_{1},\ S_{2}$ of points in ${\mathbb{B}}$ such that : if there is a bounded holomorphic function $f$ on ${\mathbb{B}}$ and a $\delta <1$ with $|{f}| \leq \delta $ on $S_{1}$ and $|{f}| \geq 1$ on $S_{2},$ then $S$ is a separated Carleson sequence. \quad \quad In one variable we prove that this condition is also necessary. Because in one variable separated Carleson sequence is equivalent to interpolating sequence, we recover a characterization of interpolating sequences just by one function.

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