{"title":"On the Weyl Multipliers for General Haar and Franklin Systems","authors":"G. Gevorkyan","doi":"10.3103/s106836232470016x","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>In the work the almost everywhere (a.e.) convergence (absolute convergence) of series by the general Haar and Franklin systems corresponding to weakly regular division of the segment <span>\\([0,1]\\)</span> are compared. It is proved that if a series by the general Haar system diverges (absolutely diverges) on a set <span>\\(E\\)</span>, then the series by the general Franklin system with the same coefficients diverges (absolutely diverges) a.e. in <span>\\(E\\)</span>. As a consequence, it is obtained that if a sequence <span>\\(\\omega_{n}\\)</span> is not a Weyl multiplier for unconditional a.e. convergence of series by the general Haar system, then it is not a Weyl multiplier for unconditional a.e. convergence of series by the general Franklin series.</p>","PeriodicalId":54854,"journal":{"name":"Journal of Contemporary Mathematical Analysis-Armenian Academy of Sciences","volume":"11 1","pages":""},"PeriodicalIF":0.3000,"publicationDate":"2024-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Contemporary Mathematical Analysis-Armenian Academy of Sciences","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.3103/s106836232470016x","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
In the work the almost everywhere (a.e.) convergence (absolute convergence) of series by the general Haar and Franklin systems corresponding to weakly regular division of the segment \([0,1]\) are compared. It is proved that if a series by the general Haar system diverges (absolutely diverges) on a set \(E\), then the series by the general Franklin system with the same coefficients diverges (absolutely diverges) a.e. in \(E\). As a consequence, it is obtained that if a sequence \(\omega_{n}\) is not a Weyl multiplier for unconditional a.e. convergence of series by the general Haar system, then it is not a Weyl multiplier for unconditional a.e. convergence of series by the general Franklin series.
Abstract In the work the almost everywhere (a.e.) convergence (absolute convergence) of series by the general Haar and Franklin systems corresponding to weakly regular division of the segment \([0,1]\) are compared.结果证明，如果一般哈尔系统的数列在一个集合 \（E\ ）上发散（绝对发散），那么具有相同系数的一般富兰克林系统的数列在 \（E\ ）内发散（绝对发散）。由此可以得出，如果一个序列 \(\omega_{n}\)不是一般哈氏系统数列无条件a.e.收敛的韦尔乘数，那么它也不是一般富兰克林数列无条件a.e.收敛的韦尔乘数。
期刊介绍：
Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences) is an outlet for research stemming from the widely acclaimed Armenian school of theory of functions, this journal today continues the traditions of that school in the area of general analysis. A very prolific group of mathematicians in Yerevan contribute to this leading mathematics journal in the following fields: real and complex analysis; approximations; boundary value problems; integral and stochastic geometry; differential equations; probability; integral equations; algebra.
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