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On reverse Markov–Nikol’skii inequalities for polynomials with restricted zeros 关于有限制零点的多项式的反向马尔可夫-尼克尔斯基不等式
IF 0.9 3区 数学
Journal of Approximation Theory Pub Date : 2024-08-30 DOI: 10.1016/j.jat.2024.106097
Mikhail A. Komarov
{"title":"On reverse Markov–Nikol’skii inequalities for polynomials with restricted zeros","authors":"Mikhail A. Komarov","doi":"10.1016/j.jat.2024.106097","DOIUrl":"10.1016/j.jat.2024.106097","url":null,"abstract":"<div><p>Let <span><math><msub><mrow><mi>Π</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> be the class of algebraic polynomials <span><math><mi>P</mi></math></span> of degree <span><math><mi>n</mi></math></span>, all of whose zeros lie on the segment <span><math><mrow><mo>[</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></math></span>. In 1995, S. P. Zhou has proved the following Turán type reverse Markov–Nikol’skii inequality: <span><math><mrow><msub><mrow><mo>‖</mo><msup><mrow><mi>P</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>‖</mo></mrow><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub><mrow><mo>[</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></mrow></msub><mo>&gt;</mo><mi>c</mi><mspace></mspace><msup><mrow><mrow><mo>(</mo><msqrt><mrow><mi>n</mi></mrow></msqrt><mo>)</mo></mrow></mrow><mrow><mn>1</mn><mo>−</mo><mn>1</mn><mo>/</mo><mi>p</mi><mo>+</mo><mn>1</mn><mo>/</mo><mi>q</mi></mrow></msup><mspace></mspace><msub><mrow><mo>‖</mo><mi>P</mi><mo>‖</mo></mrow><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>q</mi></mrow></msub><mrow><mo>[</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></mrow></msub></mrow></math></span>, <span><math><mrow><mi>P</mi><mo>∈</mo><msub><mrow><mi>Π</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></math></span>, where <span><math><mrow><mn>0</mn><mo>&lt;</mo><mi>p</mi><mo>≤</mo><mi>q</mi><mo>≤</mo><mi>∞</mi></mrow></math></span>, <span><math><mrow><mn>1</mn><mo>−</mo><mn>1</mn><mo>/</mo><mi>p</mi><mo>+</mo><mn>1</mn><mo>/</mo><mi>q</mi><mo>≥</mo><mn>0</mn></mrow></math></span>\u0000(<span><math><mrow><mi>c</mi><mo>&gt;</mo><mn>0</mn></mrow></math></span> is a constant independent of <span><math><mi>P</mi></math></span> and <span><math><mi>n</mi></math></span>). We show that Zhou’s estimate remains true in the case <span><math><mrow><mi>p</mi><mo>=</mo><mi>∞</mi></mrow></math></span>, <span><math><mrow><mi>q</mi><mo>&gt;</mo><mn>1</mn></mrow></math></span>. Some of related Turán type inequalities are also discussed.</p></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"305 ","pages":"Article 106097"},"PeriodicalIF":0.9,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142096759","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Distribution of the zeros of polynomials near the unit circle 单位圆附近多项式零点的分布
IF 0.9 3区 数学
Journal of Approximation Theory Pub Date : 2024-08-08 DOI: 10.1016/j.jat.2024.106087
Mithun Kumar Das
{"title":"Distribution of the zeros of polynomials near the unit circle","authors":"Mithun Kumar Das","doi":"10.1016/j.jat.2024.106087","DOIUrl":"10.1016/j.jat.2024.106087","url":null,"abstract":"<div><p>We estimate the number of zeros of a polynomial in <span><math><mrow><mi>ℂ</mi><mrow><mo>[</mo><mi>z</mi><mo>]</mo></mrow></mrow></math></span> within any small circular disk centered on the unit circle, which improves and comprehensively extends a result established by Borwein, Erdélyi, and Littmann in 2008. Furthermore, by combining this result with Euclidean geometry, we derive an upper bound on the number of zeros of such a polynomial within a region resembling a gear wheel. Additionally, we obtain a sharp upper bound on the annular discrepancy of such zeros near the unit circle. Our approach builds upon a modified version of the method described in Borwein et al. (2008), combined with the refined version of the best-known upper bound for angular discrepancy of zeros of polynomials.</p></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"304 ","pages":"Article 106087"},"PeriodicalIF":0.9,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141964469","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Convergence in distribution of the Bernstein–Durrmeyer kernel and pointwise convergence of a generalised operator for functions of bounded variation 伯恩斯坦-达尔迈耶核分布的收敛性和有界变化函数广义算子的点收敛性
IF 0.9 3区 数学
Journal of Approximation Theory Pub Date : 2024-08-08 DOI: 10.1016/j.jat.2024.106086
Mohammed Taariq Mowzer
{"title":"Convergence in distribution of the Bernstein–Durrmeyer kernel and pointwise convergence of a generalised operator for functions of bounded variation","authors":"Mohammed Taariq Mowzer","doi":"10.1016/j.jat.2024.106086","DOIUrl":"10.1016/j.jat.2024.106086","url":null,"abstract":"<div><p>We study the convergence of Bernstein type operators leading to two results. The first: The kernel <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> of the Bernstein–Durrmeyer operator at each point <span><math><mrow><mi>x</mi><mo>∈</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span> — that is <span><math><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mi>d</mi><mi>t</mi></mrow></math></span> — once standardised converges to the normal distribution. The second result computes the pointwise limit of a generalised Bernstein–Durrmeyer operator applied to — possibly discontinuous — functions <span><math><mi>f</mi></math></span> of bounded variation.</p></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"304 ","pages":"Article 106086"},"PeriodicalIF":0.9,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0021904524000741/pdfft?md5=3ef19eb55045a8c776031676ab20fd9e&pid=1-s2.0-S0021904524000741-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141979096","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Singular examples of the Matrix Bochner Problem 矩阵波赫纳问题的奇异实例
IF 0.9 3区 数学
Journal of Approximation Theory Pub Date : 2024-07-26 DOI: 10.1016/j.jat.2024.106082
Ignacio Bono Parisi, Inés Pacharoni
{"title":"Singular examples of the Matrix Bochner Problem","authors":"Ignacio Bono Parisi,&nbsp;Inés Pacharoni","doi":"10.1016/j.jat.2024.106082","DOIUrl":"10.1016/j.jat.2024.106082","url":null,"abstract":"<div><p>The Matrix Bochner Problem aims to classify which weight matrices have their sequence of orthogonal polynomials as eigenfunctions of a second-order differential operator. Casper and Yakimov, in <span><span>[4]</span></span>, demonstrated that, under certain hypotheses, all solutions to the Matrix Bochner Problem are noncommutative bispectral Darboux transformations of a direct sum of classical scalar weights. This paper aims to provide the first proof that there are solutions to the Matrix Bochner Problem that do not arise through a noncommutative bispectral Darboux transformation of any direct sum of classical scalar weights. This initial example could contribute to a more comprehensive understanding of the general solution to the Matrix Bochner Problem.</p></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"304 ","pages":"Article 106082"},"PeriodicalIF":0.9,"publicationDate":"2024-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141851133","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Minimal projections onto subspaces generated by sign-matrices 符号矩阵生成的子空间上的最小投影
IF 0.9 3区 数学
Journal of Approximation Theory Pub Date : 2024-07-26 DOI: 10.1016/j.jat.2024.106084
Beata Derȩgowska , Barbara Lewandowska , Grzegorz Lewicki
{"title":"Minimal projections onto subspaces generated by sign-matrices","authors":"Beata Derȩgowska ,&nbsp;Barbara Lewandowska ,&nbsp;Grzegorz Lewicki","doi":"10.1016/j.jat.2024.106084","DOIUrl":"10.1016/j.jat.2024.106084","url":null,"abstract":"<div><p>The aim of this paper is to calculate relative and absolute projection constants of certain subspaces of <span><math><msubsup><mrow><mi>l</mi></mrow><mrow><mn>1</mn></mrow><mrow><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></msubsup></math></span> and <span><math><msubsup><mrow><mi>l</mi></mrow><mrow><mi>∞</mi></mrow><mrow><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></msubsup></math></span> generated by eigenvectors of sign matrices. The main tool in our considerations is so called Chalmers–Metcalf operator (see Chalmers &amp; Metcalf (1994) and Lewicki &amp; Prophet (2021)). Also, some results from Castejon &amp; Lewicki (2014) and Castejon &amp; Lewicki (2019) will be applied.</p></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"304 ","pages":"Article 106084"},"PeriodicalIF":0.9,"publicationDate":"2024-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141846148","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On approximation by rational functions in Musielak–Orlicz spaces 论穆西拉克-奥利兹空间中有理函数的逼近
IF 0.9 3区 数学
Journal of Approximation Theory Pub Date : 2024-07-26 DOI: 10.1016/j.jat.2024.106083
Wojciech M. Kozlowski , Gianluca Vinti
{"title":"On approximation by rational functions in Musielak–Orlicz spaces","authors":"Wojciech M. Kozlowski ,&nbsp;Gianluca Vinti","doi":"10.1016/j.jat.2024.106083","DOIUrl":"10.1016/j.jat.2024.106083","url":null,"abstract":"<div><p>We consider best approximation by rational functions in Musielak–Orlicz spaces of real-valued measurable functions over the unit interval equipped with the Lebesgue measure. We prove several properties of the respective multi-value projection operator, including its semi-continuity. Our results generalise known results for Lebesgue and variable Lebesgues spaces, and can be applied to special cases including Orlicz spaces and variable Lebesgue spaces with weights. We touch upon applications to image processing.</p></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"304 ","pages":"Article 106083"},"PeriodicalIF":0.9,"publicationDate":"2024-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0021904524000716/pdfft?md5=772962309c292532c60924a31afa1fa7&pid=1-s2.0-S0021904524000716-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141847123","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The Machado–Bishop theorem in the uniform topology 统一拓扑中的马查多-毕夏普定理
IF 0.9 3区 数学
Journal of Approximation Theory Pub Date : 2024-07-26 DOI: 10.1016/j.jat.2024.106085
Deliang Chen
{"title":"The Machado–Bishop theorem in the uniform topology","authors":"Deliang Chen","doi":"10.1016/j.jat.2024.106085","DOIUrl":"10.1016/j.jat.2024.106085","url":null,"abstract":"<div><p>The Machado–Bishop theorem for weighted vector-valued functions vanishing at infinity has been extensively studied. In this paper, we give an analogue of Machado’s distance formula for bounded weighted vector-valued functions. A number of applications are given; in particular, some types of the Bishop–Stone–Weierstrass theorem for bounded vector-valued continuous spaces in the uniform topology are discussed; the splitting of <span><math><mrow><mi>C</mi><mrow><mo>(</mo><mi>I</mi><mo>×</mo><mi>J</mi><mo>,</mo><mi>X</mi><mo>⊗</mo><mi>Y</mi><mo>)</mo></mrow></mrow></math></span> as the closure of <span><math><mrow><mi>C</mi><mrow><mo>(</mo><mi>I</mi><mo>,</mo><mi>X</mi><mo>)</mo></mrow><mo>⊗</mo><mi>C</mi><mrow><mo>(</mo><mi>J</mi><mo>,</mo><mi>Y</mi><mo>)</mo></mrow></mrow></math></span> in different senses and the extension of continuous vector-valued functions are studied.</p></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"304 ","pages":"Article 106085"},"PeriodicalIF":0.9,"publicationDate":"2024-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141848755","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Remembering Peter Benjamin Borwein (May 10, 1953–August 23, 2020) 缅怀彼得-本杰明-博文（1953 年 5 月 10 日-2020 年 8 月 23 日）
IF 0.9 3区 数学
Journal of Approximation Theory Pub Date : 2024-07-06 DOI: 10.1016/j.jat.2024.106070
Tamás Erdélyi
{"title":"Remembering Peter Benjamin Borwein (May 10, 1953–August 23, 2020)","authors":"Tamás Erdélyi","doi":"10.1016/j.jat.2024.106070","DOIUrl":"10.1016/j.jat.2024.106070","url":null,"abstract":"","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"303 ","pages":"Article 106070"},"PeriodicalIF":0.9,"publicationDate":"2024-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141700410","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Peter Borwein: A philosopher’s mathematician 彼得-博文哲学家的数学家
IF 0.9 3区 数学
Journal of Approximation Theory Pub Date : 2024-07-05 DOI: 10.1016/j.jat.2024.106071
David H. Bailey
{"title":"Peter Borwein: A philosopher’s mathematician","authors":"David H. Bailey","doi":"10.1016/j.jat.2024.106071","DOIUrl":"10.1016/j.jat.2024.106071","url":null,"abstract":"","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"303 ","pages":"Article 106071"},"PeriodicalIF":0.9,"publicationDate":"2024-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141690540","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
In memory of Peter Borwein 纪念彼得-博尔文
IF 0.9 3区 数学
Journal of Approximation Theory Pub Date : 2024-07-05 DOI: 10.1016/j.jat.2024.106072
Stephen Choi
{"title":"In memory of Peter Borwein","authors":"Stephen Choi","doi":"10.1016/j.jat.2024.106072","DOIUrl":"10.1016/j.jat.2024.106072","url":null,"abstract":"","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"303 ","pages":"Article 106072"},"PeriodicalIF":0.9,"publicationDate":"2024-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0021904524000583/pdfft?md5=cbc6f03a9f2eb4bee6ad9cfdd54f3ec1&pid=1-s2.0-S0021904524000583-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141695786","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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