Journal of Approximation Theory最新文献

{"title":"Infinite-dimensional integration and L2-approximation on Hermite spaces","authors":"M. Gnewuch ,&nbsp;A. Hinrichs ,&nbsp;K. Ritter ,&nbsp;R. Rüßmann","doi":"10.1016/j.jat.2024.106027","DOIUrl":"10.1016/j.jat.2024.106027","url":null,"abstract":"<div><p>We study integration and <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-approximation of functions of infinitely many variables in the following setting: The underlying function space is the countably infinite tensor product of univariate Hermite spaces and the probability measure is the corresponding product of the standard normal distribution. The maximal domain of the functions from this tensor product space is necessarily a proper subset of the sequence space <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span>. We establish upper and lower bounds for the minimal worst case errors under general assumptions; these bounds do match for tensor products of well-studied Hermite spaces of functions with finite or with infinite smoothness. In the proofs we employ embedding results, and the upper bounds are attained constructively with the help of multivariate decomposition methods.</p></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139828571","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}

{"title":"Existence and uniqueness of s-curve segments of tensioned elastica satisfying geometric Hermite interpolation conditions","authors":"Michael J. Johnson","doi":"10.1016/j.jat.2024.106017","DOIUrl":"https://doi.org/10.1016/j.jat.2024.106017","url":null,"abstract":"<div><p>It has been recently proved that every <em>proper</em> restricted elastic spline is a stable nonlinear spline, and this yields a broad existence proof for stable nonlinear splines. When tension is included in the setup, stable nonlinear splines under tension always exist, but they do not always have the property that each piece (connecting one interpolation point to the next) is an s-curve. Being correlated with the fairness of an interpolating curve, this property is desirable and we conjecture that the framework employed successfully with restricted elastic splines will also work well with nonlinear splines under tension. Our purpose is to prove the following foundational result: Given points <span><math><mrow><msub><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>≠</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math></span>, in the plane, along with corresponding unit directions <span><math><mrow><msub><mrow><mi>d</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>d</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math></span> that satisfy <span><math><mrow><msub><mrow><mi>d</mi></mrow><mrow><mn>1</mn></mrow></msub><mi>⋅</mi><mrow><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>−</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mo>≥</mo><mn>0</mn></mrow></math></span> and <span><math><mrow><msub><mrow><mi>d</mi></mrow><mrow><mn>2</mn></mrow></msub><mi>⋅</mi><mrow><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>−</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mo>≥</mo><mn>0</mn></mrow></math></span>, there exists a unique s-curve segment of Euler–Bernoulli elastica under tension <span><math><mrow><mi>λ</mi><mo>&gt;</mo><mn>0</mn></mrow></math></span> that connects <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> to <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> with initial direction <span><math><msub><mrow><mi>d</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and terminal direction <span><math><msub><mrow><mi>d</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>.</p></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139985540","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}

{"title":"Inradius of random lemniscates","authors":"Manjunath Krishnapur ,&nbsp;Erik Lundberg ,&nbsp;Koushik Ramachandran","doi":"10.1016/j.jat.2024.106018","DOIUrl":"10.1016/j.jat.2024.106018","url":null,"abstract":"<div><p>A classically studied geometric property associated to a complex polynomial <span><math><mi>p</mi></math></span> is the inradius (the radius of the largest inscribed disk) of its (filled) lemniscate <span><math><mrow><mi>Λ</mi><mo>≔</mo><mrow><mo>{</mo><mi>z</mi><mo>∈</mo><mi>ℂ</mi><mo>:</mo><mrow><mo>|</mo><mi>p</mi><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>|</mo></mrow><mo>&lt;</mo><mn>1</mn><mo>}</mo></mrow></mrow></math></span>.</p><p>In this paper, we study the lemniscate inradius when the defining polynomial <span><math><mi>p</mi></math></span> is random, namely, with the zeros of <span><math><mi>p</mi></math></span> sampled independently from a compactly supported probability measure <span><math><mi>μ</mi></math></span>. If the negative set of the logarithmic potential <span><math><msub><mrow><mi>U</mi></mrow><mrow><mi>μ</mi></mrow></msub></math></span> generated by <span><math><mi>μ</mi></math></span> is non-empty, then the inradius is bounded from below by a positive constant with overwhelming probability (as the degree <span><math><mi>n</mi></math></span> of <span><math><mi>p</mi></math></span> tends to infinity). Moreover, the inradius has a deterministic limit if the negative set of <span><math><msub><mrow><mi>U</mi></mrow><mrow><mi>μ</mi></mrow></msub></math></span> additionally contains the support of <span><math><mi>μ</mi></math></span>.</p><p>We also provide conditions on <span><math><mi>μ</mi></math></span> guaranteeing that the lemniscate is contained in a union of <span><math><mi>n</mi></math></span> exponentially small disks with overwhelming probability. This leads to a partial solution to a (deterministic) problem concerning the area of lemniscates posed by Erdös, Herzog, and Piranian.</p><p>On the other hand, when the zeros are sampled independently and uniformly from the unit circle, then we show that the inradius converges in distribution to a random variable taking values in <span><math><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo>)</mo></mrow></math></span>.</p><p>We also consider the characteristic polynomial of a Ginibre random matrix whose lemniscate we show is close to the unit disk with overwhelming probability.</p></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139679171","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}

{"title":"Infinite-dimensional integration and L2-approximation on Hermite spaces","authors":"M. Gnewuch, A. Hinrichs, K. Ritter, R. Rüßmann","doi":"10.1016/j.jat.2024.106027","DOIUrl":"https://doi.org/10.1016/j.jat.2024.106027","url":null,"abstract":"","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139888602","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}

{"title":"Estimates of linear expressions through factorization","authors":"Ali Hasan Ali ,&nbsp;Zsolt Páles","doi":"10.1016/j.jat.2024.106019","DOIUrl":"10.1016/j.jat.2024.106019","url":null,"abstract":"<div><p>The aim of this paper is to establish various factorization results and then to derive estimates for linear functionals through the use of a generalized Taylor theorem. Additionally, several error bounds are established including applications to the trapezoidal rule as well as to a Simpson formula-type rule.</p></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0021904524000054/pdfft?md5=acddbc03e340f31280d60e4dcc812a41&pid=1-s2.0-S0021904524000054-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139555798","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}

{"title":"Multivariate polynomial splines on generalized oranges","authors":"Maritza Sirvent ,&nbsp;Tatyana Sorokina ,&nbsp;Nelly Villamizar ,&nbsp;Beihui Yuan","doi":"10.1016/j.jat.2024.106016","DOIUrl":"10.1016/j.jat.2024.106016","url":null,"abstract":"<div><p>We consider spaces of multivariate splines defined on a particular type of simplicial partitions that we call <em>(generalized) oranges</em>. Such partitions are composed of a finite number of maximal faces with exactly one shared <em>medial</em> face. We reduce the problem of finding the dimension of splines on oranges to computing dimensions of splines on simpler, lower-dimensional partitions that we call <em>projected oranges</em>. We use both algebraic and Bernstein–Bézier tools.</p></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0021904524000029/pdfft?md5=4b8bba34d42f748261b9923cce3ba6be&pid=1-s2.0-S0021904524000029-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139496860","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}

{"title":"Localization for random CMV matrices","authors":"Xiaowen Zhu","doi":"10.1016/j.jat.2023.106008","DOIUrl":"10.1016/j.jat.2023.106008","url":null,"abstract":"<div><p>We prove Anderson localization (AL) and dynamical localization in expectation (EDL, also known as strong dynamical localization) for random CMV matrices for arbitrary distribution of i.i.d. Verblunsky coefficients.</p></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139396463","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}

{"title":"Comonotone approximation of periodic functions","authors":"D. Leviatan ,&nbsp;M.V. Shchehlov ,&nbsp;I.O. Shevchuk","doi":"10.1016/j.jat.2024.106015","DOIUrl":"10.1016/j.jat.2024.106015","url":null,"abstract":"<div><p>Let <span><math><mover><mrow><mi>C</mi></mrow><mrow><mo>˜</mo></mrow></mover></math></span> be the space of continuous <span><math><mrow><mn>2</mn><mi>π</mi></mrow></math></span>-periodic functions <span><math><mi>f</mi></math></span>, endowed with the uniform norm <span><math><mrow><mo>‖</mo><mi>f</mi><mo>‖</mo><mo>≔</mo><msub><mrow><mo>max</mo></mrow><mrow><mi>x</mi><mo>∈</mo><mi>R</mi></mrow></msub><mrow><mo>|</mo><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>|</mo></mrow></mrow></math></span>, and denote by <span><math><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mi>m</mi></mrow></msub><mrow><mo>(</mo><mi>f</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow></mrow></math></span>, the <span><math><mi>m</mi></math></span>th modulus of smoothness of <span><math><mi>f</mi></math></span>. Denote by <span><math><msup><mrow><mover><mrow><mi>C</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>r</mi></mrow></msup></math></span>, the subspace of <span><math><mi>r</mi></math></span><span> times continuously differentiable functions </span><span><math><mrow><mi>f</mi><mo>∈</mo><mover><mrow><mi>C</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow></math></span>, and let <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span><span>, be the set of trigonometric polynomials </span><span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> of degree <span><math><mrow><mo>&lt;</mo><mi>n</mi></mrow></math></span>. If <span><math><mrow><mi>f</mi><mo>∈</mo><msup><mrow><mover><mrow><mi>C</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>r</mi></mrow></msup></mrow></math></span>, has <span><math><mrow><mn>2</mn><mi>s</mi></mrow></math></span>, <span><math><mrow><mi>s</mi><mo>≥</mo><mn>1</mn></mrow></math></span><span>, extremal points in </span><span><math><mrow><mo>(</mo><mo>−</mo><mi>π</mi><mo>,</mo><mi>π</mi><mo>]</mo></mrow></math></span>, denote by <span><math><mrow><msubsup><mrow><mi>E</mi></mrow><mrow><mi>n</mi></mrow><mrow><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></msubsup><mrow><mo>(</mo><mi>f</mi><mo>)</mo></mrow><mo>≔</mo><munder><mrow><mo>inf</mo></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>∈</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>:</mo><msup><mrow><mi>f</mi></mrow><mrow><mo>′</mo></mrow></msup><msubsup><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>′</mo></mrow></msubsup><mo>≥</mo><mn>0</mn></mrow></munder><mo>‖</mo><mi>f</mi><mo>−</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>‖</mo><mo>,</mo></mrow></math></span> the error of its best comonotone approximation. We prove, that if <span><math><mrow><mi>f</mi><mo>∈</mo><msup><mrow><mover><mrow><mi>C</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>r</mi></mrow></msup></mrow></math></span>, then for either <span><math><mrow><mi>m</mi><mo>=</mo><mn>1</mn></mrow></math></span>, or <span><math><mrow><mi>m</mi><mo>=</mo><mn>2<","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139101849","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}

{"title":"Polynomial approximation on disjoint segments and amplification of approximation","authors":"Yu. Malykhin ,&nbsp;K. Ryutin","doi":"10.1016/j.jat.2023.106010","DOIUrl":"10.1016/j.jat.2023.106010","url":null,"abstract":"<div><p><span>We construct explicit easily implementable polynomial approximations of sufficiently high accuracy for locally constant functions on the union of disjoint segments (see </span><span>(1)</span><span><span>). This problem has important applications in several areas of numerical analysis, complexity theory, </span>quantum algorithms, etc. The one, most relevant for us, is the amplification of approximation method: it allows to construct approximations of higher degree </span><span><math><mi>M</mi></math></span><span> and better accuracy from the approximations of degree </span><span><math><mi>m</mi></math></span>.</p></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139102000","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}

{"title":"Onesided, intertwining, positive and copositive polynomial approximation with interpolatory constraints","authors":"German Dzyubenko ,&nbsp;Kirill A. Kopotun","doi":"10.1016/j.jat.2023.106012","DOIUrl":"10.1016/j.jat.2023.106012","url":null,"abstract":"<div><p>Given <span><math><mrow><mi>k</mi><mo>∈</mo><mi>N</mi></mrow></math></span>, a nonnegative function <span><math><mrow><mi>f</mi><mo>∈</mo><msup><mrow><mi>C</mi></mrow><mrow><mi>r</mi></mrow></msup><mrow><mo>[</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>]</mo></mrow></mrow></math></span>, <span><math><mrow><mi>r</mi><mo>≥</mo><mn>0</mn></mrow></math></span>, an arbitrary finite collection of points <span><math><mrow><msub><mrow><mrow><mo>{</mo><mrow><msub><mrow><mi>α</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mo>}</mo></mrow></mrow><mrow><mi>i</mi><mo>∈</mo><mi>J</mi></mrow></msub><mo>⊂</mo><mrow><mo>[</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>]</mo></mrow></mrow></math></span><span>, and a corresponding collection of nonnegative integers </span><span><math><msub><mrow><mrow><mo>{</mo><mrow><msub><mrow><mi>m</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mo>}</mo></mrow></mrow><mrow><mi>i</mi><mo>∈</mo><mi>J</mi></mrow></msub></math></span> with <span><math><mrow><mn>0</mn><mo>≤</mo><msub><mrow><mi>m</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>≤</mo><mi>r</mi></mrow></math></span>, <span><math><mrow><mi>i</mi><mo>∈</mo><mi>J</mi></mrow></math></span>, is it true that, for sufficiently large <span><math><mrow><mi>n</mi><mo>∈</mo><mi>N</mi></mrow></math></span>, there exists a polynomial <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> of degree <span><math><mi>n</mi></math></span> such that</p><p>(i) <span><math><mrow><mrow><mo>|</mo><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>−</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>|</mo></mrow><mo>≤</mo><mi>c</mi><msubsup><mrow><mi>ρ</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>r</mi></mrow></msubsup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><msub><mrow><mi>ω</mi></mrow><mrow><mi>k</mi></mrow></msub><mrow><mo>(</mo><msup><mrow><mi>f</mi></mrow><mrow><mrow><mo>(</mo><mi>r</mi><mo>)</mo></mrow></mrow></msup><mo>,</mo><msub><mrow><mi>ρ</mi></mrow><mrow><mi>n</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>;</mo><mrow><mo>[</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>]</mo></mrow><mo>)</mo></mrow></mrow></math></span>, <span><math><mrow><mi>x</mi><mo>∈</mo><mrow><mo>[</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>]</mo></mrow></mrow></math></span>, where <span><math><mrow><msub><mrow><mi>ρ</mi></mrow><mrow><mi>n</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>≔</mo><msup><mrow><mi>n</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><msqrt><mrow><mn>1</mn><mo>−</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msqrt><mo>+</mo><msup><mrow><mi>n</mi></mrow><mrow><mo>−</mo><mn>2</mn></mrow></msup></mrow></math></span> and <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> is the classical <span><math><mi>k</mi></math></span>th modulus of smoothness.</p><p>(ii) <span><math><mrow><msup><mrow><mi>P</mi></mrow><mrow><mrow","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139093119","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}