{"title":"Almost Periodic Functions: Their Limit Sets and Various Applications","authors":"Lev Sakhnovich","doi":"10.1007/s40315-023-00515-2","DOIUrl":null,"url":null,"abstract":"<p>In the present paper, we introduce and study the limit sets of the almost periodic functions <i>f</i>: <span>\\({{\\mathbb {R}}}\\rightarrow {{\\mathbb {C}}}\\)</span>. It is interesting, that <span>\\(r=\\inf |f(x)|\\)</span> and <span>\\(R=\\sup |f(x)|\\)</span> may be expressed in exact form. In particular, the formula for <i>r</i> coincides with the well known partition problem formula. We show that the ring <span>\\(r\\le |z|\\le R\\)</span> is the limit set of the almost periodic function <i>f</i>(<i>x</i>) (under some natural conditions on <i>f</i>). We obtain interesting applications to the partition problem in number theory and to trigonometric series theory. We extend classical results for periodic functions (estimation of Fourier coefficients and Bernstein’s theorem about absolute convergence of trigonometric series) to almost periodic functions. Using our results on almost periodic functions, we propose a new approach to interesting problems of the stability of motion. An application of our function theoretic results to the spectral problems in operator theory is studied as well. The results and corresponding computations are illustrated by several figures.</p>","PeriodicalId":49088,"journal":{"name":"Computational Methods and Function Theory","volume":"116 1","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2023-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Computational Methods and Function Theory","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s40315-023-00515-2","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
In the present paper, we introduce and study the limit sets of the almost periodic functions f: \({{\mathbb {R}}}\rightarrow {{\mathbb {C}}}\). It is interesting, that \(r=\inf |f(x)|\) and \(R=\sup |f(x)|\) may be expressed in exact form. In particular, the formula for r coincides with the well known partition problem formula. We show that the ring \(r\le |z|\le R\) is the limit set of the almost periodic function f(x) (under some natural conditions on f). We obtain interesting applications to the partition problem in number theory and to trigonometric series theory. We extend classical results for periodic functions (estimation of Fourier coefficients and Bernstein’s theorem about absolute convergence of trigonometric series) to almost periodic functions. Using our results on almost periodic functions, we propose a new approach to interesting problems of the stability of motion. An application of our function theoretic results to the spectral problems in operator theory is studied as well. The results and corresponding computations are illustrated by several figures.
期刊介绍:
CMFT is an international mathematics journal which publishes carefully selected original research papers in complex analysis (in a broad sense), and on applications or computational methods related to complex analysis. Survey articles of high standard and current interest can be considered for publication as well.