论哈代核作为再生核

Pub Date : 2022-06-17 DOI:10.4153/S0008439522000406
J. Oliva-Maza
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引用次数: 0

摘要

Hardy核是在Hilbertian空间(如$L^2(\mathbb R^+)$或$H^2(\mathbb C^+)$上定义积分算子的一个有用工具。这些核需要一个代数的$L^1$ -结构,在这项工作中使用它来研究这些算子的值域空间作为核希尔伯特空间的再现。我们得到了它们的再现核,在L^2(\mathbb R^+)$的情况下,它也是哈代核。在$H^2(\mathbb C^+)$情形中,复制核是由Hardy核的全纯扩展给出的。这里给出的其他结果是Paley-Wiener型定理,以及与单侧希尔伯特变换的联系。
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On Hardy kernels as reproducing kernels
Abstract Hardy kernels are a useful tool to define integral operators on Hilbertian spaces like $L^2(\mathbb R^+)$ or $H^2(\mathbb C^+)$ . These kernels entail an algebraic $L^1$ -structure which is used in this work to study the range spaces of those operators as reproducing kernel Hilbert spaces. We obtain their reproducing kernels, which in the $L^2(\mathbb R^+)$ case turn out to be Hardy kernels as well. In the $H^2(\mathbb C^+)$ scenario, the reproducing kernels are given by holomorphic extensions of Hardy kernels. Other results presented here are theorems of Paley–Wiener type, and a connection with one-sided Hilbert transforms.
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