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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Interpolating sequences in polydisks

Author: Eric P. Kronstadt
Journal: Trans. Amer. Math. Soc. 199 (1974), 369-398
MSC: Primary 32E25; Secondary 46J10
MathSciNet review: 0417451
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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.

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Keywords: Interpolating sequence, uniformly separated sequence, general interpolating sequence, nontangential wedge, uniform algebra valued functions
Article copyright: © Copyright 1974 American Mathematical Society

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