{"title":"Uniqueness for Spherically Convergent Multiple Trigonometric Series","authors":"J. Ash","doi":"10.1201/9780429123610-7","DOIUrl":"https://doi.org/10.1201/9780429123610-7","url":null,"abstract":"In 1870 Cantor proved that representation of a function of one variable by a trigonometric series can be done in only one way. In 1996 Bourgain proved the same thing for spherical convergence and multiple trigonometric series. His proof involves injecting a lot of new ideas into the theory of uniqueness. We give here an exposition of Bourgain’s proof, specialized to the case of dimension 2. Our exposition includes a fairly general method for nding maximal elements without resorting to the Axiom of Choice.","PeriodicalId":139586,"journal":{"name":"Handbook of Analytic-Computational Methods in Applied Mathematics","volume":"269 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133512724","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}