{"title":"Cauchy-Szegö 核的加权组合及其均值导数的最佳近似值","authors":"Viktor V. Savchuk, Maryna V. Savchuk","doi":"arxiv-2409.10833","DOIUrl":null,"url":null,"abstract":"In this paper, we study an extremal problem concerning best approximation in\nthe Hardy space $H^1$ on the unit disk $\\mathbb D$. Specifically, we consider\nweighted combinations of the Cauchy-Szeg\\\"o kernel and its derivative,\nparametrized by an inner function $\\varphi$ and a complex number $\\lambda$, and\nprovide explicit formula of the best approximation $e_{\\varphi,z}(\\lambda)$ by\nthe subspace $H^1_0$. We also describe the extremal functions associated with\nthis approximation.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"29 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Best approximations for the weighted combination of the Cauchy--Szegö kernel and its derivative in the mean\",\"authors\":\"Viktor V. Savchuk, Maryna V. Savchuk\",\"doi\":\"arxiv-2409.10833\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In this paper, we study an extremal problem concerning best approximation in\\nthe Hardy space $H^1$ on the unit disk $\\\\mathbb D$. Specifically, we consider\\nweighted combinations of the Cauchy-Szeg\\\\\\\"o kernel and its derivative,\\nparametrized by an inner function $\\\\varphi$ and a complex number $\\\\lambda$, and\\nprovide explicit formula of the best approximation $e_{\\\\varphi,z}(\\\\lambda)$ by\\nthe subspace $H^1_0$. We also describe the extremal functions associated with\\nthis approximation.\",\"PeriodicalId\":501142,\"journal\":{\"name\":\"arXiv - MATH - Complex Variables\",\"volume\":\"29 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-17\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Complex Variables\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2409.10833\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Complex Variables","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.10833","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Best approximations for the weighted combination of the Cauchy--Szegö kernel and its derivative in the mean
In this paper, we study an extremal problem concerning best approximation in
the Hardy space $H^1$ on the unit disk $\mathbb D$. Specifically, we consider
weighted combinations of the Cauchy-Szeg\"o kernel and its derivative,
parametrized by an inner function $\varphi$ and a complex number $\lambda$, and
provide explicit formula of the best approximation $e_{\varphi,z}(\lambda)$ by
the subspace $H^1_0$. We also describe the extremal functions associated with
this approximation.