{"title":"On moments and symmetrical sequences","authors":"Jiten Ahuja, Ricardo Estrada","doi":"10.1016/j.indag.2024.04.008","DOIUrl":null,"url":null,"abstract":"<div><p>In this article we consider questions related to the behavior of the moments <span><math><mrow><msub><mrow><mi>M</mi></mrow><mrow><mi>m</mi></mrow></msub><mfenced><mrow><mfenced><mrow><msub><mrow><mi>z</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></mfenced></mrow></mfenced></mrow></math></span> when the indices are restricted to specific subsequences of integers, such as the even or odd moments. If <span><math><mrow><mi>n</mi><mo>≥</mo><mn>2</mn></mrow></math></span> we introduce the notion of symmetrical series of order <span><math><mrow><mi>n</mi><mo>,</mo></mrow></math></span> showing that if <span><math><mrow><mfenced><mrow><msub><mrow><mi>z</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></mfenced><mspace></mspace></mrow></math></span> is symmetrical then <span><math><mrow><msub><mrow><mi>M</mi></mrow><mrow><mi>m</mi></mrow></msub><mfenced><mrow><mfenced><mrow><msub><mrow><mi>z</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></mfenced></mrow></mfenced><mo>=</mo><mn>0</mn></mrow></math></span> whenever <span><math><mrow><mi>n</mi><mo>∤</mo><mi>m</mi><mo>;</mo></mrow></math></span> in particular, the odd moments of a symmetrical series of order 2 vanish. We prove that when <span><math><mrow><mfenced><mrow><msub><mrow><mi>z</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></mfenced><mo>∈</mo><msup><mrow><mi>l</mi></mrow><mrow><mi>p</mi></mrow></msup></mrow></math></span> for some <span><math><mi>p</mi></math></span> then several results characterizing the sequence from its moments hold. We show, in particular, that if <span><math><mrow><msub><mrow><mi>M</mi></mrow><mrow><mi>m</mi></mrow></msub><mfenced><mrow><mfenced><mrow><msub><mrow><mi>z</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></mfenced></mrow></mfenced><mo>=</mo><mn>0</mn></mrow></math></span> whenever <span><math><mrow><mi>n</mi><mo>∤</mo><mi>m</mi></mrow></math></span> then <span><math><mfenced><mrow><msub><mrow><mi>z</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></mfenced></math></span> is a rearrangement of a symmetrical series of order <span><math><mrow><mi>n</mi><mo>.</mo></mrow></math></span> We then construct examples of sequences whose moments vanish with required density. Lastly, we construct counterexamples of several of the results valid in the <span><math><msup><mrow><mi>l</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span> case if we allow the moment series to be all <em>conditionally convergent</em>. We show that for each <em>arbitrary</em> sequence of real numbers <span><math><msubsup><mrow><mfenced><mrow><msub><mrow><mi>μ</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow></mfenced></mrow><mrow><mi>m</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>∞</mi></mrow></msubsup></math></span> there are real sequences <span><math><msubsup><mrow><mfenced><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></mfenced></mrow><mrow><mi>j</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>∞</mi></mrow></msubsup></math></span> such that <span><span><span><math><mrow><munderover><mrow><mo>∑</mo></mrow><mrow><mi>j</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>∞</mi></mrow></munderover><msubsup><mrow><mi>u</mi></mrow><mrow><mi>j</mi></mrow><mrow><mn>2</mn><mi>m</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo>=</mo><msub><mrow><mi>μ</mi></mrow><mrow><mi>m</mi></mrow></msub><mspace></mspace><mspace></mspace><mspace></mspace><mspace></mspace><mi>m</mi><mo>≥</mo><mn>0</mn><mspace></mspace><mo>.</mo></mrow></math></span></span></span></p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"35 3","pages":"Pages 584-594"},"PeriodicalIF":0.5000,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Indagationes Mathematicae-New Series","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0019357724000405","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 this article we consider questions related to the behavior of the moments when the indices are restricted to specific subsequences of integers, such as the even or odd moments. If we introduce the notion of symmetrical series of order showing that if is symmetrical then whenever in particular, the odd moments of a symmetrical series of order 2 vanish. We prove that when for some then several results characterizing the sequence from its moments hold. We show, in particular, that if whenever then is a rearrangement of a symmetrical series of order We then construct examples of sequences whose moments vanish with required density. Lastly, we construct counterexamples of several of the results valid in the case if we allow the moment series to be all conditionally convergent. We show that for each arbitrary sequence of real numbers there are real sequences such that
期刊介绍:
Indagationes Mathematicae is a peer-reviewed international journal for the Mathematical Sciences of the Royal Dutch Mathematical Society. The journal aims at the publication of original mathematical research papers of high quality and of interest to a large segment of the mathematics community. The journal also welcomes the submission of review papers of high quality.