Gunther Leobacher , Friedrich Pillichshammer , Adrian Ebert
{"title":"Tractability of L2-approximation and integration in weighted Hermite spaces of finite smoothness","authors":"Gunther Leobacher , Friedrich Pillichshammer , Adrian Ebert","doi":"10.1016/j.jco.2023.101768","DOIUrl":null,"url":null,"abstract":"<div><p>In this paper we consider integration and <span><math><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-approximation for functions over <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>s</mi></mrow></msup></math></span> from weighted Hermite spaces. The first part of the paper is devoted to a comparison of several weighted Hermite spaces that appear in literature, which is interesting on its own. Then we study tractability of the integration and <span><math><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-approximation problem for the introduced Hermite spaces, which describes the growth rate of the information complexity when the error threshold <em>ε</em> tends to 0 and the problem dimension <em>s</em> grows to infinity. Our main results are characterizations of tractability in terms of the involved weights, which model the importance of the successive coordinate directions for functions from the weighted Hermite spaces.</p></div>","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0885064X23000377","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
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Abstract
In this paper we consider integration and -approximation for functions over from weighted Hermite spaces. The first part of the paper is devoted to a comparison of several weighted Hermite spaces that appear in literature, which is interesting on its own. Then we study tractability of the integration and -approximation problem for the introduced Hermite spaces, which describes the growth rate of the information complexity when the error threshold ε tends to 0 and the problem dimension s grows to infinity. Our main results are characterizations of tractability in terms of the involved weights, which model the importance of the successive coordinate directions for functions from the weighted Hermite spaces.
期刊介绍：
Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance.
Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.
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