Journal of Approximation Theory最新文献

{"title":"On sharp heat kernel estimates in the context of Fourier–Dini expansions","authors":"Bartosz Langowski ,&nbsp;Adam Nowak","doi":"10.1016/j.jat.2024.106103","DOIUrl":"10.1016/j.jat.2024.106103","url":null,"abstract":"<div><div>We prove sharp estimates of the heat kernel associated with Fourier–Dini expansions on <span><math><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></math></span> equipped with Lebesgue measure and the Neumann condition imposed on the right endpoint. Then we give several applications of this result including sharp bounds for the corresponding Poisson and potential kernels, sharp mapping properties of the maximal heat semigroup and potential operators and boundary convergence of the Fourier–Dini semigroup.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"304 ","pages":"Article 106103"},"PeriodicalIF":0.9,"publicationDate":"2024-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142424156","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":"Translation-based completeness on compact intervals","authors":"Lukas Liehr","doi":"10.1016/j.jat.2024.106104","DOIUrl":"10.1016/j.jat.2024.106104","url":null,"abstract":"<div><div>Given a compact interval <span><math><mrow><mi>I</mi><mo>⊆</mo><mi>R</mi></mrow></math></span>, and a function <span><math><mi>f</mi></math></span> that is a product of a nonzero polynomial with a Gaussian, it will be shown that the translates <span><math><mrow><mo>{</mo><mi>f</mi><mrow><mo>(</mo><mi>⋅</mi><mo>−</mo><mi>λ</mi><mo>)</mo></mrow><mo>:</mo><mi>λ</mi><mo>∈</mo><mi>Λ</mi><mo>}</mo></mrow></math></span> are complete in <span><math><mrow><mi>C</mi><mrow><mo>(</mo><mi>I</mi><mo>)</mo></mrow></mrow></math></span> if and only if the series of reciprocals of <span><math><mi>Λ</mi></math></span> diverges. This extends a theorem in [R. A. Zalik, Trans. Amer. Math. Soc. 243, 299–308]. An additional characterization is obtained when <span><math><mi>Λ</mi></math></span> is an arithmetic progression, and the generator <span><math><mi>f</mi></math></span> constitutes a linear combination of translates of a function with sufficiently fast decay.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"305 ","pages":"Article 106104"},"PeriodicalIF":0.9,"publicationDate":"2024-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142421197","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":"Monotonicity of zeros of derivatives of Bessel functions","authors":"Dimitar K. Dimitrov,&nbsp;Yen Chi Lun","doi":"10.1016/j.jat.2024.106102","DOIUrl":"10.1016/j.jat.2024.106102","url":null,"abstract":"<div><div>Recently Baricz et al., 2018 and Baricz and Singh 2018 gave two different proofs of the fact that the zeros of the <span><math><mi>n</mi></math></span>th derivative of the Bessel function of the first kind <span><math><mrow><msub><mrow><mi>J</mi></mrow><mrow><mi>ν</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span> are all real when <span><math><mrow><mi>ν</mi><mo>&gt;</mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow></math></span>. We provide a third alternative proof. The authors of Baricz et al., 2018 conjectured that, for every <span><math><mrow><mi>n</mi><mo>∈</mo><mi>N</mi></mrow></math></span>, the positive zeros of <span><math><mrow><msubsup><mrow><mi>J</mi></mrow><mrow><mi>ν</mi></mrow><mrow><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></msubsup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span> are increasing functions of the parameter <span><math><mi>ν</mi></math></span>, for <span><math><mrow><mi>ν</mi><mo>∈</mo><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow></mrow></math></span>. We provide two apparently distinct proofs of the conjecture.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"305 ","pages":"Article 106102"},"PeriodicalIF":0.9,"publicationDate":"2024-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142432541","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":"On Bernstein- and Marcinkiewicz-type inequalities on multivariate Cα-domains","authors":"Feng Dai ,&nbsp;András Kroó ,&nbsp;Andriy Prymak","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":"305 ","pages":"Article 106101"},"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}

{"title":"Lower bounds for piecewise polynomial approximations of oscillatory functions","authors":"Jeffrey Galkowski","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":"305 ","pages":"Article 106100"},"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}

{"title":"Function recovery on manifolds using scattered data","authors":"David Krieg ,&nbsp;Mathias Sonnleitner","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":"305 ","pages":"Article 106098"},"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}

{"title":"Dual spaces vs. Haar measures of polynomial hypergroups","authors":"Stefan Kahler ,&nbsp;Ryszard Szwarc","doi":"10.1016/j.jat.2024.106099","DOIUrl":"10.1016/j.jat.2024.106099","url":null,"abstract":"<div><div>Many symmetric orthogonal polynomials <span><math><msub><mrow><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></mrow><mrow><mi>n</mi><mo>∈</mo><msub><mrow><mi>N</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></msub></math></span> induce a hypergroup structure on <span><math><msub><mrow><mi>N</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>. The Haar measure is the counting measure weighted with <span><math><mrow><mi>h</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>≔</mo><mn>1</mn><mo>/</mo><msub><mrow><mo>∫</mo></mrow><mrow><mi>R</mi></mrow></msub><mspace></mspace><msubsup><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mspace></mspace><mi>d</mi><mi>μ</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>≥</mo><mn>1</mn></mrow></math></span>, where <span><math><mi>μ</mi></math></span> denotes the orthogonalization measure. We observed that many naturally occurring examples satisfy the remarkable property <span><math><mrow><mi>h</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>≥</mo><mn>2</mn><mspace></mspace><mrow><mo>(</mo><mi>n</mi><mo>∈</mo><mi>N</mi><mo>)</mo></mrow></mrow></math></span>. We give sufficient criteria and particularly show that <span><math><mrow><mi>h</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>≥</mo><mn>2</mn><mspace></mspace><mrow><mo>(</mo><mi>n</mi><mo>∈</mo><mi>N</mi><mo>)</mo></mrow></mrow></math></span> if the (Hermitian) dual space <span><math><mover><mrow><msub><mrow><mi>N</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow><mrow><mo>̂</mo></mrow></mover></math></span> equals the full interval <span><math><mrow><mo>[</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></math></span>, which is fulfilled by an abundance of examples. We also study the role of nonnegative linearization of products (and of the resulting harmonic and functional analysis). Moreover, we construct two example types with <span><math><mrow><mi>h</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow><mo>&lt;</mo><mn>2</mn></mrow></math></span>. To our knowledge, these are the first such examples. The first type is based on Karlin–McGregor polynomials, and <span><math><mover><mrow><msub><mrow><mi>N</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow><mrow><mo>̂</mo></mrow></mover></math></span> consists of two intervals and can be chosen “maximal” in some sense; <span><math><mi>h</mi></math></span> is of quadratic growth. The second type relies on hypergroups of strong compact type; <span><math><mi>h</mi></math></span> grows exponentially, and <span><math><mover><mrow><msub><mrow><mi>N</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow><mrow><mo>̂</mo></mrow></mover></math></span> is discrete.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"305 ","pages":"Article 106099"},"PeriodicalIF":0.9,"publicationDate":"2024-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142421236","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":"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}

{"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}

{"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}