{"title":"论哈代核作为再生核","authors":"J. Oliva-Maza","doi":"10.4153/S0008439522000406","DOIUrl":null,"url":null,"abstract":"Abstract Hardy kernels are a useful tool to define integral operators on Hilbertian spaces like \n$L^2(\\mathbb R^+)$\n or \n$H^2(\\mathbb C^+)$\n . These kernels entail an algebraic \n$L^1$\n -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 \n$L^2(\\mathbb R^+)$\n case turn out to be Hardy kernels as well. In the \n$H^2(\\mathbb C^+)$\n 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.","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On Hardy kernels as reproducing kernels\",\"authors\":\"J. Oliva-Maza\",\"doi\":\"10.4153/S0008439522000406\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Abstract Hardy kernels are a useful tool to define integral operators on Hilbertian spaces like \\n$L^2(\\\\mathbb R^+)$\\n or \\n$H^2(\\\\mathbb C^+)$\\n . These kernels entail an algebraic \\n$L^1$\\n -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 \\n$L^2(\\\\mathbb R^+)$\\n case turn out to be Hardy kernels as well. In the \\n$H^2(\\\\mathbb C^+)$\\n 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.\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2022-06-17\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.4153/S0008439522000406\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4153/S0008439522000406","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
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.