Representing Measures and Hardy Spaces for the Infinite Polydisk Algebra
Representing Measures and Hardy Spaces for the Infinite Polydisk Algebra
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DOI：
10.1112/plms/s3-53.1.112
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发表时间：
1986-07
影响因子：
1.8
通讯作者：
B. Cole;T. Gamelin
中科院分区：
文献类型：
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作者：
B. Cole;T. Gamelin
本文研究了由坐标函数z1，z2，.生成的无限多圆盘上的一致代数的表示测度和詹森测度。设σ是无限环面T ∞上的Haar测度，T ∞是无限多圆盘的区别边界.对于固定的值域1 <p，< ∞，证明了点λ ∈ Δ∞在LP（σ）中有表示测度当且仅当λ ∈l2.对于原点的一类表示测度（包括Haar测度σ），以及对于0 <p< ∞范围内的固定p，一个相关的结果是，在λ处的点赋值在Lp-范数下连续当且仅当λ ∈ Δ∞ <$l2。在这种情况下，H中的函数对应于域Δ∞ l2 inl 2上的解析函数。沿着同样的思路，证明了H ∞（σ）与H（Δ∞ <$l2）和H ∞（B）是等距同构的，其中B是序列空间c 0的开单位球。
Representing measures and Jensen measures are studied for the uniform algebra on the infinite polydiskgenerated by the coordinate functions z1z2,…. Let σ be the Haar measure on the infinite torusT∞, which is the distinguished boundary of the infinite polydisk. For fixedpin the range 1 <p, < ∞, it is shown that a point ζ ∈ Δ∞ has a representing measure inLP(σ) if and only if ζ ∈l2. A related result for a class of representing measures for the origin, including the Haar measure σ, and for fixed p in the range 0 <p< ∞, is that the point evaluation at ζ is continuous in theLp-norm if and only if ζ ∈ Δ∞ ∩l2. In this case the functions inHpare shown to correspond to analytic functions on the domain Δ∞l2inl2. Along these same lines, it is shown thatH∞ (σ) is isometrically isomorphic toH(Δ∞ ∩l2), and also toH∞(B), whereBis the open unit ball of the sequence spacec0.