{"title":"Characterization of Holomorphic Functions by Zero Spherical Means","authors":"N. P. Volchkova, Vit. V. Volchkov","doi":"10.1134/s1055134424030076","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We continue to study the holomorphy problem for functions whose contour integrals over\ncircles vanish. We consider the case in which a function <span>\\(f \\)</span> is defined on a deleted ball <span>\\(\\mathcal {D} \\)</span> in <span>\\(\\mathbb {C}^n\\)</span>\n(without its center) and integrate over all spheres of two fixed radii inside <span>\\(\\mathcal {D} \\)</span>. For <span>\\(f\\in C^{\\infty }(\\mathcal {D}) \\)</span>, we find conditions on the radii and size of\n<span>\\(\\mathcal {D} \\)</span> implying that <span>\\(f \\)</span> is a holomorphic function. We also show that these\nconditions cannot be weakened in the general case.\n</p>","PeriodicalId":39997,"journal":{"name":"Siberian Advances in Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Siberian Advances in Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1134/s1055134424030076","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
We continue to study the holomorphy problem for functions whose contour integrals over
circles vanish. We consider the case in which a function \(f \) is defined on a deleted ball \(\mathcal {D} \) in \(\mathbb {C}^n\)
(without its center) and integrate over all spheres of two fixed radii inside \(\mathcal {D} \). For \(f\in C^{\infty }(\mathcal {D}) \), we find conditions on the radii and size of
\(\mathcal {D} \) implying that \(f \) is a holomorphic function. We also show that these
conditions cannot be weakened in the general case.
期刊介绍：
Siberian Advances in Mathematics is a journal that publishes articles on fundamental and applied mathematics. It covers a broad spectrum of subjects: algebra and logic, real and complex analysis, functional analysis, differential equations, mathematical physics, geometry and topology, probability and mathematical statistics, mathematical cybernetics, mathematical economics, mathematical problems of geophysics and tomography, numerical methods, and optimization theory.
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