{"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":55280,"journal":{"name":"Canadian Mathematical Bulletin-Bulletin Canadien De Mathematiques","volume":"66 1","pages":"428 - 442"},"PeriodicalIF":0.5000,"publicationDate":"2022-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Canadian Mathematical Bulletin-Bulletin Canadien De Mathematiques","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4153/S0008439522000406","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
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.
期刊介绍:
The Canadian Mathematical Bulletin was established in 1958 to publish original, high-quality research papers in all branches of mathematics and to accommodate the growing demand for shorter research papers. The Bulletin is a companion publication to the Canadian Journal of Mathematics that publishes longer papers. New research papers are published continuously online and collated into print issues four times each year.
To be submitted to the Bulletin, papers should be at most 18 pages long and may be written in English or in French. Longer papers should be submitted to the Canadian Journal of Mathematics.
Fondé en 1958, le Bulletin canadien de mathématiques (BCM) publie des articles d’avant-garde et de grande qualité dans toutes les branches des mathématiques, de même que pour répondre à la demande croissante d’articles scientifiques plus brefs. Le BCM se veut une publication complémentaire au Journal canadien de mathématiques, qui publie de longs articles. En ligne, il propose constamment de nouveaux articles de recherche, puis les réunit dans des numéros imprimés quatre fois par année.
Les textes présentés au BCM doivent compter au plus 18 pages et être rédigés en anglais ou en français. C’est le Journal canadien de mathématiques qui reçoit les articles plus longs.