Interpolating sequences in polydisks

Abstract:

Let ${D^n}$ be the unit polydisk in ${{\mathbf {C}}^n}$, $A$ be a uniform algebra, ${H^\infty }({D^n},A)$, the space of bounded analytic $A$ valued functions on ${D^n}$, ${l^\infty }A$, the space of bounded sequences of elements in $A$. A sequence, $S = \{ {a_i}\} _{i = 1}^\infty \subset {D^n}$ will be called an interpolating sequence with respect to $A$ if the map $T:{H^\infty }({D^n},A) \to {l^\infty }A$ given by $T(f) = \{ f({a_i})\} _{i = 1}^\infty$ is surjective. In 1958, L. Carleson showed that for $n = 1,S$ is interpolating w.r.t. ${\mathbf {C}}$ iff $S$ satisfies a certain zero-one interpolation property called uniform separation. We generalize this result to cases where $n > 1$ and $A \ne {\mathbf {C}}$. In particular, we show that if $S \subset {D^n}$ is uniformly separated and $S \subset {W_1} \times {W_2} \times \cdots \times {W_n}$ (where each ${W_j}$ is a region in $D$ lying between two circular arcs which intersect twice on the boundary of $D$) then $S$ is an interpolating sequence w.r.t. any uniform algebra. If $S \subset {D^n}$ is uniformly separated and $S \subset D \times {W_2} \times \cdots \times {W_n}$ then $S$ is interpolating w.r.t. ${\mathbf {C}}$. Other examples and generalizations of interpolating sequences are discussed.