{"title":"关于Zahariuta猜想和Kolmogorov问题","authors":"Stéphanie Nivoche","doi":"10.1016/S0764-4442(01)02042-0","DOIUrl":null,"url":null,"abstract":"<div><p>We prove the Zahariuta's conjecture, which itself solves a Kolmogorov's problem on the <em>ε</em>-entropy of classes of analytic functions. For a given holomorphically convex compact subset <em>K</em> in a bounded pseudoconvex domain <em>D</em> in <span><math><mtext>C</mtext><msup><mi></mi><mn>n</mn></msup></math></span>, the Zahariuta's conjecture consists in approximating uniformly on any compact subset of <em>D</em>⧹<em>K</em>, the relative extremal function <em>u</em><sub><em>K</em>,<em>D</em></sub> by a sequence of pluricomplex Green functions on <em>D</em> with logarithmic poles in the compact set <em>K</em>.</p></div>","PeriodicalId":100300,"journal":{"name":"Comptes Rendus de l'Académie des Sciences - Series I - Mathematics","volume":"333 9","pages":"Pages 839-843"},"PeriodicalIF":0.0000,"publicationDate":"2001-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0764-4442(01)02042-0","citationCount":"5","resultStr":"{\"title\":\"Sur une conjecture de Zahariuta et un problème de Kolmogorov\",\"authors\":\"Stéphanie Nivoche\",\"doi\":\"10.1016/S0764-4442(01)02042-0\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We prove the Zahariuta's conjecture, which itself solves a Kolmogorov's problem on the <em>ε</em>-entropy of classes of analytic functions. For a given holomorphically convex compact subset <em>K</em> in a bounded pseudoconvex domain <em>D</em> in <span><math><mtext>C</mtext><msup><mi></mi><mn>n</mn></msup></math></span>, the Zahariuta's conjecture consists in approximating uniformly on any compact subset of <em>D</em>⧹<em>K</em>, the relative extremal function <em>u</em><sub><em>K</em>,<em>D</em></sub> by a sequence of pluricomplex Green functions on <em>D</em> with logarithmic poles in the compact set <em>K</em>.</p></div>\",\"PeriodicalId\":100300,\"journal\":{\"name\":\"Comptes Rendus de l'Académie des Sciences - Series I - Mathematics\",\"volume\":\"333 9\",\"pages\":\"Pages 839-843\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2001-11-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://sci-hub-pdf.com/10.1016/S0764-4442(01)02042-0\",\"citationCount\":\"5\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Comptes Rendus de l'Académie des Sciences - Series I - Mathematics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0764444201020420\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Comptes Rendus de l'Académie des Sciences - Series I - Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0764444201020420","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Sur une conjecture de Zahariuta et un problème de Kolmogorov
We prove the Zahariuta's conjecture, which itself solves a Kolmogorov's problem on the ε-entropy of classes of analytic functions. For a given holomorphically convex compact subset K in a bounded pseudoconvex domain D in , the Zahariuta's conjecture consists in approximating uniformly on any compact subset of D⧹K, the relative extremal function uK,D by a sequence of pluricomplex Green functions on D with logarithmic poles in the compact set K.
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