{"title":"On the Fourier Coefficients of Powers of a Finite Blaschke Product","authors":"Alexander Borichev, Karine Fouchet, Rachid Zarouf","doi":"10.1093/imrn/rnae199","DOIUrl":null,"url":null,"abstract":"Given a finite Blaschke product $B$ we prove asymptotically sharp estimates on the $\\ell ^{\\infty }$-norm of the sequence of the Fourier coefficients of $B^{n}$ as $n$ tends to $\\infty $. This norm decays as $n^{-1/N}$ for some $N\\ge 3$. Furthermore, for every $N\\ge 3$, we produce explicitly a finite Blaschke product $B$ with decay $n^{-1/N}$. As an application we construct a sequence of $n\\times n$ invertible matrices $T$ with arbitrary spectrum in the unit disk and such that the quantity $|\\det{T}|\\cdot \\|T^{-1}\\|\\cdot \\|T\\|^{1-n}$ grows as a power of $n$. This is motivated by Schäffer’s question on norms of inverses.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":"42 1","pages":""},"PeriodicalIF":0.9000,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"International Mathematics Research Notices","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1093/imrn/rnae199","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Given a finite Blaschke product $B$ we prove asymptotically sharp estimates on the $\ell ^{\infty }$-norm of the sequence of the Fourier coefficients of $B^{n}$ as $n$ tends to $\infty $. This norm decays as $n^{-1/N}$ for some $N\ge 3$. Furthermore, for every $N\ge 3$, we produce explicitly a finite Blaschke product $B$ with decay $n^{-1/N}$. As an application we construct a sequence of $n\times n$ invertible matrices $T$ with arbitrary spectrum in the unit disk and such that the quantity $|\det{T}|\cdot \|T^{-1}\|\cdot \|T\|^{1-n}$ grows as a power of $n$. This is motivated by Schäffer’s question on norms of inverses.
期刊介绍:
International Mathematics Research Notices provides very fast publication of research articles of high current interest in all areas of mathematics. All articles are fully refereed and are judged by their contribution to advancing the state of the science of mathematics.