{"title":"整个函数的bernstein型表征","authors":"O. Dovgoshey, Jürgen Prestin, I. Shevchuk","doi":"10.15407/dopovidi2023.01.010","DOIUrl":null,"url":null,"abstract":"Let ε be the set of all entire functions on the complex plane C. Let us consider the class XE of all complex Banach spaces X such that X ⊇ ε . For (X, ⎥⎥ ⋅ ⎥⎥)∈XE and g ∈X we write En, X (g ) = inf {⎥⎥ g − p⎥⎥: p∈Πn }, where Πn is the set of all polynomials with degree at most n. We describe all X ∈XE for which the relation lim n→∞ (En, X( g ))1/n = 0 holds if and only if g ∈ ε.","PeriodicalId":20898,"journal":{"name":"Reports of the National Academy of Sciences of Ukraine","volume":"4 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2023-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Bernstein-type characterization of entire functions\",\"authors\":\"O. Dovgoshey, Jürgen Prestin, I. Shevchuk\",\"doi\":\"10.15407/dopovidi2023.01.010\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let ε be the set of all entire functions on the complex plane C. Let us consider the class XE of all complex Banach spaces X such that X ⊇ ε . For (X, ⎥⎥ ⋅ ⎥⎥)∈XE and g ∈X we write En, X (g ) = inf {⎥⎥ g − p⎥⎥: p∈Πn }, where Πn is the set of all polynomials with degree at most n. We describe all X ∈XE for which the relation lim n→∞ (En, X( g ))1/n = 0 holds if and only if g ∈ ε.\",\"PeriodicalId\":20898,\"journal\":{\"name\":\"Reports of the National Academy of Sciences of Ukraine\",\"volume\":\"4 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2023-03-09\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Reports of the National Academy of Sciences of Ukraine\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.15407/dopovidi2023.01.010\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Reports of the National Academy of Sciences of Ukraine","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.15407/dopovidi2023.01.010","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
摘要
设ε为复平面c上所有完整函数的集合，我们考虑所有复巴拿赫空间X的类XE，使得X。为(X,⎥⎥⋅⎥⎥)∈XE和g∈X我们写En, X (g) = inf{⎥⎥g−p⎥⎥:p∈Πn},其中Πn的所有多项式程度最多n。我们描述所有X∈XE的关系lim n→∞(En, X (g)) 1 / n = 0成立当且仅当g∈ε。
Bernstein-type characterization of entire functions
Let ε be the set of all entire functions on the complex plane C. Let us consider the class XE of all complex Banach spaces X such that X ⊇ ε . For (X, ⎥⎥ ⋅ ⎥⎥)∈XE and g ∈X we write En, X (g ) = inf {⎥⎥ g − p⎥⎥: p∈Πn }, where Πn is the set of all polynomials with degree at most n. We describe all X ∈XE for which the relation lim n→∞ (En, X( g ))1/n = 0 holds if and only if g ∈ ε.
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