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On Bernstein- and Marcinkiewicz-type inequalities on multivariate Cα-domains 论多变量 Cα 域上的伯恩斯坦和马钦凯维奇型不等式
IF 0.9 3区 数学
Journal of Approximation Theory Pub Date : 2024-09-16 DOI: 10.1016/j.jat.2024.106101
{"title":"On Bernstein- and Marcinkiewicz-type inequalities on multivariate Cα-domains","authors":"","doi":"10.1016/j.jat.2024.106101","DOIUrl":"10.1016/j.jat.2024.106101","url":null,"abstract":"<div><p>We prove new Bernstein and Markov type inequalities in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span> spaces associated with the normal and the tangential derivatives on the boundary of a general compact <span><math><msup><mrow><mi>C</mi></mrow><mrow><mi>α</mi></mrow></msup></math></span>-domain with <span><math><mrow><mn>1</mn><mo>≤</mo><mi>α</mi><mo>≤</mo><mn>2</mn></mrow></math></span>. These estimates are also applied to establish Marcinkiewicz type inequalities for discretization of <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span> norms of algebraic polynomials on <span><math><msup><mrow><mi>C</mi></mrow><mrow><mi>α</mi></mrow></msup></math></span>-domains with asymptotically optimal number of function samples used.</p></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142270714","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
Lower bounds for piecewise polynomial approximations of oscillatory functions 振荡函数的片断多项式近似值下限
IF 0.9 3区 数学
Journal of Approximation Theory Pub Date : 2024-09-14 DOI: 10.1016/j.jat.2024.106100
{"title":"Lower bounds for piecewise polynomial approximations of oscillatory functions","authors":"","doi":"10.1016/j.jat.2024.106100","DOIUrl":"10.1016/j.jat.2024.106100","url":null,"abstract":"<div><p>We prove lower bounds on the error incurred when approximating any oscillating function using piecewise polynomial spaces. The estimates are explicit in the polynomial degree and have optimal dependence on the meshwidth and frequency when the polynomial degree is fixed. These lower bounds, for example, apply when approximating solutions to Helmholtz plane wave scattering problem.</p></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0021904524000881/pdfft?md5=fb33e23c82eb14bbcd3a20a9e7b11759&pid=1-s2.0-S0021904524000881-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142270713","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
Function recovery on manifolds using scattered data 利用分散数据恢复流形上的函数
IF 0.9 3区 数学
Journal of Approximation Theory Pub Date : 2024-09-13 DOI: 10.1016/j.jat.2024.106098
{"title":"Function recovery on manifolds using scattered data","authors":"","doi":"10.1016/j.jat.2024.106098","DOIUrl":"10.1016/j.jat.2024.106098","url":null,"abstract":"<div><p>We consider the task of recovering a Sobolev function on a connected compact Riemannian manifold <span><math><mi>M</mi></math></span> when given a sample on a finite point set. We prove that the quality of the sample is given by the <span><math><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>γ</mi></mrow></msub><mrow><mo>(</mo><mi>M</mi><mo>)</mo></mrow></mrow></math></span>-average of the geodesic distance to the point set and determine the value of <span><math><mrow><mi>γ</mi><mo>∈</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>∞</mi><mo>]</mo></mrow></mrow></math></span>. This extends our findings on bounded convex domains [IMA J. Numer. Anal., 2024]. As a byproduct, we prove the optimal rate of convergence of the <span><math><mi>n</mi></math></span>th minimal worst case error for <span><math><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>q</mi></mrow></msub><mrow><mo>(</mo><mi>M</mi><mo>)</mo></mrow></mrow></math></span>-approximation for all <span><math><mrow><mn>1</mn><mo>≤</mo><mi>q</mi><mo>≤</mo><mi>∞</mi></mrow></math></span>.</p><p>Further, a limit theorem for moments of the average distance to a set consisting of i.i.d. uniform points is proven. This yields that a random sample is asymptotically as good as an optimal sample in precisely those cases with <span><math><mrow><mi>γ</mi><mo>&lt;</mo><mi>∞</mi></mrow></math></span>. In particular, we obtain that cubature formulas with random nodes are asymptotically as good as optimal cubature formulas if the weights are chosen correctly. This closes a logarithmic gap left open by Ehler, Gräf and Oates [Stat. Comput., 29:1203-1214, 2019].</p></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0021904524000868/pdfft?md5=fe139e66c2cbd25bf59dda36950e8234&pid=1-s2.0-S0021904524000868-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142239392","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
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
{"title":"On reverse Markov–Nikol’skii inequalities for polynomials with restricted zeros","authors":"","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":null,"pages":null},"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
{"title":"Distribution of the zeros of polynomials near the unit circle","authors":"","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":null,"pages":null},"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
{"title":"Convergence in distribution of the Bernstein–Durrmeyer kernel and pointwise convergence of a generalised operator for functions of bounded variation","authors":"","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":null,"pages":null},"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
{"title":"Singular examples of the Matrix Bochner Problem","authors":"","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":null,"pages":null},"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
{"title":"Minimal projections onto subspaces generated by sign-matrices","authors":"","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":null,"pages":null},"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
{"title":"On approximation by rational functions in Musielak–Orlicz spaces","authors":"","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":null,"pages":null},"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
{"title":"The Machado–Bishop theorem in the uniform topology","authors":"","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":null,"pages":null},"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
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