On the New Generalized Hahn Sequence Space h d Q3 Mathematics Abstract and Applied Analysis Pub Date : 2022-09-16 DOI:10.1155/2022/6832559 Orhan Tuǧ, E. Malkowsky, B. Hazarika, Taja Yaying 求助PDF {"title":"On the New Generalized Hahn Sequence Space <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M1\">\n <msubsup>\n <mrow>\n <mi>h</mi>\n </mrow>\n <mrow>\n <mi>d</mi","authors":"Orhan Tuǧ, E. Malkowsky, B. Hazarika, Taja Yaying","doi":"10.1155/2022/6832559","DOIUrl":null,"url":null,"abstract":"<jats:p>In this article, we define the new generalized Hahn sequence space <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M2\">\n <msubsup>\n <mrow>\n <mi>h</mi>\n </mrow>\n <mrow>\n <mi>d</mi>\n </mrow>\n <mrow>\n <mi>p</mi>\n </mrow>\n </msubsup>\n </math>\n </jats:inline-formula>, where <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M3\">\n <mi>d</mi>\n <mo>=</mo>\n <msubsup>\n <mrow>\n <mfenced open=\"(\" close=\")\">\n <mrow>\n <msub>\n <mrow>\n <mi>d</mi>\n </mrow>\n <mrow>\n <mi>k</mi>\n </mrow>\n </msub>\n </mrow>\n </mfenced>\n </mrow>\n <mrow>\n <mi>k</mi>\n <mo>=</mo>\n <mn>1</mn>\n </mrow>\n <mrow>\n <mo>∞</mo>\n </mrow>\n </msubsup>\n </math>\n </jats:inline-formula> is monotonically increasing sequence with <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M4\">\n <msub>\n <mrow>\n <mi>d</mi>\n </mrow>\n <mrow>\n <mi>k</mi>\n </mrow>\n </msub>\n <mo>≠</mo>\n <mn>0</mn>\n </math>\n </jats:inline-formula> for all <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M5\">\n <mi>k</mi>\n <mo>∈</mo>\n <mi>ℕ</mi>\n </math>\n </jats:inline-formula>, and <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M6\">\n <mn>1</mn>\n <mo><</mo>\n <mi>p</mi>\n <mo><</mo>\n <mo>∞</mo>\n </math>\n </jats:inline-formula>. Then, we prove some topological properties and calculate the <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M7\">\n <mi>α</mi>\n <mo>−</mo>\n </math>\n </jats:inline-formula>, <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M8\">\n <mi>β</mi>\n <mo>−</mo>\n </math>\n </jats:inline-formula>, and <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M9\">\n <mi>γ</mi>\n <mo>−</mo>\n </math>\n </jats:inline-formula>duals of <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M10\">\n <msubsup>\n <mrow>\n <mi>h</mi>\n </mrow>\n <mrow>\n <mi>d</mi>\n </mrow>\n <mrow>\n <mi>p</mi>\n </mrow>\n </msubsup>\n </math>\n </jats:inline-formula>. Furthermore, we characterize the new matrix classes <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M11\">\n <mfenced open=\"(\" close=\")\">\n <mrow>\n <msub>\n <mrow>\n <mi>h</mi>\n </mrow>\n <mrow>\n <mi>d</mi>\n </mrow>\n </msub>\n <mo>,</mo>\n <mi>λ</mi>\n </mrow>\n </mfenced>\n </math>\n </jats:inline-formula>, where <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M12\">\n <mi>λ</mi>\n <mo>=</mo>\n <mfenced open=\"{\" close=\"}\">\n <mrow>\n <mi>b</mi>\n <mi>v</mi>\n <mo>,</mo>\n <mi>b</mi>\n <msub>\n <mrow>\n <mi>v</mi>\n </mrow>\n <mrow>\n <mi>p</mi>\n </mrow>\n </msub>\n <mo>,</mo>\n <mi>b</mi>\n <msub>\n <mrow>\n <mi>v</mi>\n </mrow>\n <mrow>\n <mo>∞</mo>\n </mrow>\n </msub>\n <mo>,</mo>\n <mi>b</mi>\n <mi>s</mi>\n <mo>,</mo>\n <mi>c</mi>\n <mi>s</mi>\n <mo>,</mo>\n </mrow>\n </mfenced>\n </math>\n </jats:inline-formula>, and <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M13\">\n <mfenced open=\"(\" close=\")\">\n <mrow>\n <mi>μ</mi>\n <mo>,</mo>\n <msub>\n <mrow>\n <mi>h</mi>\n </mrow>\n <mrow>\n <mi>d</mi>\n </mrow>\n </msub>\n </mrow>\n </mfenced>\n </math>\n </jats:inline-formula>, where <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M14\">\n <mi>μ</mi>\n <mo>=</mo>\n <mfenced open=\"{\" close=\"}\">\n <mrow>\n <mi>b</mi>\n <mi>v</mi>\n <mo>,</mo>\n <mi>b</mi>\n <msub>\n <mrow>\n <mi>v</mi>\n </mrow>\n <mrow>\n <mn>0</mn>\n </mrow>\n </msub>\n <mo>,</mo>\n <mi>b</mi>\n <mi>s</mi>\n <mo>,</mo>\n <mi>c</mi>\n <msub>\n <mrow>\n <mi>s</mi>\n </mrow>\n <mrow>\n <mn>0</mn>\n </mrow>\n </msub>\n <mo>,</mo>\n <mi>c</mi>\n <mi>s</mi>\n </mrow>\n </mfenced>\n </math>\n </jats:inline-formula>. In the last section, we prove the necessary and sufficient conditions of the matrix transformations from <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M15\">\n <msubsup>\n <mrow>\n <mi>h</mi>\n </mrow>\n <mrow>\n <mi>d</mi>\n </mrow>\n <mrow>\n <mi>p</mi>\n </mrow>\n </msubsup>\n </math>\n </jats:inline-formula> into <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M16\">\n <mi>λ</mi>\n ","PeriodicalId":7061,"journal":{"name":"Abstract and Applied Analysis","volume":" ","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2022-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Abstract and Applied Analysis","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1155/2022/6832559","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"Mathematics","Score":null,"Total":0} 引用次数: 1 引用 批量引用 Abstract In this article, we define the new generalized Hahn sequence space h d p , where d = d k k = 1 ∞ is monotonically increasing sequence with d k ≠ 0 for all k ∈ ℕ , and 1 < p < ∞ . Then, we prove some topological properties and calculate the α − , β − , and γ − duals of h d p . Furthermore, we characterize the new matrix classes h d , λ , where λ = b v , b v p , b v ∞ , b s , c s , , and μ , h d , where μ = b v , b v 0 , b s , c s 0 , c s . In the last section, we prove the necessary and sufficient conditions of the matrix transformations from h d p into λ 分享 查看原文 本刊更多论文 关于新的广义Hahn序列空间h d 在本文中，我们定义了新的广义Hahn序列空间hdp，其中d=dk k=1∞是具有dk的单调递增序列对于所有k∈ℕ , 和1P∞。然后，我们证明了一些拓扑性质，并计算了α−，h d的γ-对偶p 本文章由计算机程序翻译，如有差异，请以英文原文为准。 求助全文 约1分钟内获得全文 求助全文 来源期刊 Abstract and Applied Analysis 数学-数学 CiteScore 2.30 自引率 0.00% 发文量 36 审稿时长 3.5 months 期刊介绍： Abstract and Applied Analysis is a mathematical journal devoted exclusively to the publication of high-quality research papers in the fields of abstract and applied analysis. Emphasis is placed on important developments in classical analysis, linear and nonlinear functional analysis, ordinary and partial differential equations, optimization theory, and control theory. Abstract and Applied Analysis supports the publication of original material involving the complete solution of significant problems in the above disciplines. 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Abstract

In this article, we define the new generalized Hahn sequence space $h_d^p$ , where $d={\left(d_k\right)}_{k=1}^{\infty }$ is monotonically increasing sequence with $d_k\ne 0$ for all $k\in \mathbb{N}$ , and $1<p<\infty$ . Then, we prove some topological properties and calculate the $\alpha -$ , $\beta -$ , and $\gamma -$ duals of $h_d^p$ . Furthermore, we characterize the new matrix classes $\left(h_d,\lambda \right)$ , where $\lambda =\left\{bv,b{v}_p,b{v}_{\infty },bs,cs,\right\}$ , and $\left(\mu ,h_d\right)$ , where $\mu =\left\{bv,b{v}_0,bs,c{s}_0,cs\right\}$ . In the last section, we prove the necessary and sufficient conditions of the matrix transformations from $h_d^p$ into $\lambda$