Journal of Approximation Theory最新文献

{"title":"Chebyshev polynomials corresponding to a vanishing weight","authors":"Alex Bergman,&nbsp;Olof Rubin","doi":"10.1016/j.jat.2024.106048","DOIUrl":"https://doi.org/10.1016/j.jat.2024.106048","url":null,"abstract":"<div><p>We consider weighted Chebyshev polynomials on the unit circle corresponding to a weight of the form <span><math><msup><mrow><mrow><mo>(</mo><mi>z</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></mrow><mrow><mi>s</mi></mrow></msup></math></span> where <span><math><mrow><mi>s</mi><mo>&gt;</mo><mn>0</mn></mrow></math></span>. For integer values of <span><math><mi>s</mi></math></span> this corresponds to prescribing a zero of the polynomial on the boundary. As such, we extend findings of Lachance et al. (1979), to non-integer <span><math><mi>s</mi></math></span>. Using this generalisation, we are able to relate Chebyshev polynomials on lemniscates and other, more established, categories of Chebyshev polynomials. An essential part of our proof involves the broadening of the Erdős–Lax inequality to encompass powers of polynomials. We believe that this particular result holds significance in its own right.</p></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0021904524000340/pdfft?md5=69221809242b1dccb0aa329cd8cfc72b&pid=1-s2.0-S0021904524000340-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140900862","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":"Kolmogorov widths of an intersection of a family of balls in a mixed norm","authors":"A.A. Vasil’eva","doi":"10.1016/j.jat.2024.106046","DOIUrl":"https://doi.org/10.1016/j.jat.2024.106046","url":null,"abstract":"<div><p>In this paper, order estimates for the Kolmogorov <span><math><mi>n</mi></math></span>-widths of an intersection of a family of balls in a mixed norm in the space <span><math><msubsup><mrow><mi>l</mi></mrow><mrow><mi>q</mi><mo>,</mo><mi>σ</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>k</mi></mrow></msubsup></math></span> with <span><math><mrow><mn>2</mn><mo>⩽</mo><mi>q</mi><mo>,</mo><mspace></mspace><mi>σ</mi><mo>&lt;</mo><mi>∞</mi></mrow></math></span>, <span><math><mrow><mi>n</mi><mo>⩽</mo><mi>m</mi><mi>k</mi><mo>/</mo><mn>2</mn></mrow></math></span> are obtained.</p></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140900850","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":"Some aspects of the Bergman and Hardy spaces associated with a class of generalized analytic functions","authors":"Zhongkai Li ,&nbsp;Haihua Wei","doi":"10.1016/j.jat.2024.106044","DOIUrl":"https://doi.org/10.1016/j.jat.2024.106044","url":null,"abstract":"<div><p>For <span><math><mrow><mi>λ</mi><mo>≥</mo><mn>0</mn></mrow></math></span>, a <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> function <span><math><mi>f</mi></math></span> defined on the unit disk <span><math><mi>D</mi></math></span> is said to be <span><math><mi>λ</mi></math></span>-analytic if <span><math><mrow><msub><mrow><mi>D</mi></mrow><mrow><mover><mrow><mi>z</mi></mrow><mrow><mo>̄</mo></mrow></mover></mrow></msub><mi>f</mi><mo>=</mo><mn>0</mn></mrow></math></span>, where <span><math><msub><mrow><mi>D</mi></mrow><mrow><mover><mrow><mi>z</mi></mrow><mrow><mo>̄</mo></mrow></mover></mrow></msub></math></span> is the (complex) Dunkl operator given by <span><math><mrow><msub><mrow><mi>D</mi></mrow><mrow><mover><mrow><mi>z</mi></mrow><mrow><mo>̄</mo></mrow></mover></mrow></msub><mi>f</mi><mo>=</mo><msub><mrow><mi>∂</mi></mrow><mrow><mover><mrow><mi>z</mi></mrow><mrow><mo>̄</mo></mrow></mover></mrow></msub><mi>f</mi><mo>−</mo><mi>λ</mi><mrow><mo>(</mo><mi>f</mi><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>−</mo><mi>f</mi><mrow><mo>(</mo><mover><mrow><mi>z</mi></mrow><mrow><mo>̄</mo></mrow></mover><mo>)</mo></mrow><mo>)</mo></mrow><mo>/</mo><mrow><mo>(</mo><mi>z</mi><mo>−</mo><mover><mrow><mi>z</mi></mrow><mrow><mo>̄</mo></mrow></mover><mo>)</mo></mrow></mrow></math></span>. The aim of the paper is to study several problems on the associated Bergman spaces <span><math><mrow><msubsup><mrow><mi>A</mi></mrow><mrow><mi>λ</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mrow><mo>(</mo><mi>D</mi><mo>)</mo></mrow></mrow></math></span> and Hardy spaces <span><math><mrow><msubsup><mrow><mi>H</mi></mrow><mrow><mi>λ</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mrow><mo>(</mo><mi>D</mi><mo>)</mo></mrow></mrow></math></span> for <span><math><mrow><mi>p</mi><mo>≥</mo><mn>2</mn><mi>λ</mi><mo>/</mo><mrow><mo>(</mo><mn>2</mn><mi>λ</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span>, such as boundedness of the Bergman projection, growth of functions, density, completeness, and the dual spaces of <span><math><mrow><msubsup><mrow><mi>A</mi></mrow><mrow><mi>λ</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mrow><mo>(</mo><mi>D</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><msubsup><mrow><mi>H</mi></mrow><mrow><mi>λ</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mrow><mo>(</mo><mi>D</mi><mo>)</mo></mrow></mrow></math></span>, and characterization and interpolation of <span><math><mrow><msubsup><mrow><mi>A</mi></mrow><mrow><mi>λ</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mrow><mo>(</mo><mi>D</mi><mo>)</mo></mrow></mrow></math></span>.</p></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140894810","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":"The alternating simultaneous Halpern–Lions–Wittmann–Bauschke algorithm for finding the best approximation pair for two disjoint intersections of convex sets","authors":"Yair Censor,&nbsp;Rafiq Mansour ,&nbsp;Daniel Reem","doi":"10.1016/j.jat.2024.106045","DOIUrl":"https://doi.org/10.1016/j.jat.2024.106045","url":null,"abstract":"<div><p>Given two nonempty and disjoint intersections of closed and convex subsets, we look for a best approximation pair relative to them, i.e., a pair of points, one in each intersection, attaining the minimum distance between the disjoint intersections. We propose an iterative process based on projections onto the subsets which generate the intersections. The process is inspired by the Halpern–Lions–Wittmann–Bauschke algorithm and the classical alternating process of Cheney and Goldstein, and its advantage is that there is no need to project onto the intersections themselves, a task which can be rather demanding. We prove that under certain conditions the two interlaced subsequences converge to a best approximation pair. These conditions hold, in particular, when the space is Euclidean and the subsets which generate the intersections are compact and strictly convex. Our result extends the one of Aharoni, Censor and Jiang [“Finding a best approximation pair of points for two polyhedra”, Computational Optimization and Applications 71 (2018), 509–23] who considered the case of finite-dimensional polyhedra.</p></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141067416","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":"Complex spherical designs from group orbits","authors":"Mozhgan Mohammadpour,&nbsp;Shayne Waldron","doi":"10.1016/j.jat.2024.106047","DOIUrl":"https://doi.org/10.1016/j.jat.2024.106047","url":null,"abstract":"<div><p>We consider the general question of when all orbits under the unitary action of a finite group give a complex spherical design. Those orbits which have large stabilisers are then good candidates for being optimal complex spherical designs. This is done by developing the general theory of complex designs and associated (harmonic) Molien series for group actions. As an application, we give explicit constructions of some putatively optimal real and complex spherical <span><math><mi>t</mi></math></span>-designs.</p></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0021904524000339/pdfft?md5=219ecf58a623a8cc1c9ad8954bcd36ab&pid=1-s2.0-S0021904524000339-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141083799","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":"Spectral decomposition of H1(μ) and Poincaré inequality on a compact interval — Application to kernel quadrature","authors":"Olivier Roustant ,&nbsp;Nora Lüthen ,&nbsp;Fabrice Gamboa","doi":"10.1016/j.jat.2024.106041","DOIUrl":"https://doi.org/10.1016/j.jat.2024.106041","url":null,"abstract":"<div><p>Motivated by uncertainty quantification of complex systems, we aim at finding quadrature formulas of the form <span><math><mrow><msubsup><mrow><mo>∫</mo></mrow><mrow><mi>a</mi></mrow><mrow><mi>b</mi></mrow></msubsup><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mi>d</mi><mi>μ</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></msubsup><msub><mrow><mi>w</mi></mrow><mrow><mi>i</mi></mrow></msub><mi>f</mi><mrow><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span> where <span><math><mi>f</mi></math></span> belongs to <span><math><mrow><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup><mrow><mo>(</mo><mi>μ</mi><mo>)</mo></mrow></mrow></math></span>. Here, <span><math><mi>μ</mi></math></span> belongs to a class of continuous probability distributions on <span><math><mrow><mrow><mo>[</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>]</mo></mrow><mo>⊂</mo><mi>R</mi></mrow></math></span> and <span><math><mrow><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></msubsup><msub><mrow><mi>w</mi></mrow><mrow><mi>i</mi></mrow></msub><msub><mrow><mi>δ</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msub></mrow></math></span> is a discrete probability distribution on <span><math><mrow><mo>[</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>]</mo></mrow></math></span>. We show that <span><math><mrow><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup><mrow><mo>(</mo><mi>μ</mi><mo>)</mo></mrow></mrow></math></span> is a reproducing kernel Hilbert space with a continuous kernel <span><math><mi>K</mi></math></span>, which allows to reformulate the quadrature question as a kernel (or Bayesian) quadrature problem. Although <span><math><mi>K</mi></math></span> has not an easy closed form in general, we establish a correspondence between its spectral decomposition and the one associated to Poincaré inequalities, whose common eigenfunctions form a <span><math><mi>T</mi></math></span>-system (Karlin and Studden, 1966). The quadrature problem can then be solved in the finite-dimensional proxy space spanned by the first eigenfunctions. The solution is given by a generalized Gaussian quadrature, which we call Poincaré quadrature.</p><p>We derive several results for the Poincaré quadrature weights and the associated worst-case error. When <span><math><mi>μ</mi></math></span> is the uniform distribution, the results are explicit: the Poincaré quadrature is equivalent to the midpoint (rectangle) quadrature rule. Its nodes coincide with the zeros of an eigenfunction and the worst-case error scales as <span><math><mrow><mfrac><mrow><mi>b</mi><mo>−</mo><mi>a</mi></mrow><mrow><mn>2</mn><msqrt><mrow><mn>3</mn></mrow></msqrt></mrow></mfrac><msup><mrow><mi>n</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></math></span> for lar","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140645440","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":"Log-concavity of B-splines","authors":"Michael S. Floater","doi":"10.1016/j.jat.2024.106042","DOIUrl":"https://doi.org/10.1016/j.jat.2024.106042","url":null,"abstract":"<div><p>Curry and Schoenberg showed that a B-spline is log-concave in its support by applying Brunn’s theorem to a simplex. In this note we provide an alternative, ‘analytic’ proof of the log-concave property using only recursion formulas for B-splines and their first and second derivatives.</p></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0021904524000285/pdfft?md5=e7cbce1cee37c3e76009eab70b3d59a1&pid=1-s2.0-S0021904524000285-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140540730","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":"Weighted estimates for Hermite pseudo-multipliers with rough symbols","authors":"Fu Ken Ly","doi":"10.1016/j.jat.2024.106043","DOIUrl":"https://doi.org/10.1016/j.jat.2024.106043","url":null,"abstract":"<div><p>We introduce a class of rough symbols for pseudo-multipliers for Hermite expansions and obtain <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span> and weighted <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span> estimates. These symbols generalise the class of rough symbols introduced by Kenig–Staubach.</p></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0021904524000297/pdfft?md5=fa08028486fb2973a12d793b81945cd7&pid=1-s2.0-S0021904524000297-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140550901","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":"Nonlinear approximation of high-dimensional anisotropic analytic functions","authors":"Diane Guignard ,&nbsp;Peter Jantsch","doi":"10.1016/j.jat.2024.106040","DOIUrl":"https://doi.org/10.1016/j.jat.2024.106040","url":null,"abstract":"<div><p>Motivated by nonlinear approximation results for classes of parametric partial differential equations (PDEs), we seek to better understand so-called library approximations to analytic functions of countably infinite number of variables. Rather than approximating a function of interest by a single space, a library approximation uses a collection of spaces and the best space may be chosen for any point in the domain. In the setting of this paper, we use a specific library which consists of local Taylor approximations on sufficiently small rectangular subdomains of the (rescaled) parameter domain <span><math><mrow><mi>Y</mi><mo>≔</mo><msup><mrow><mrow><mo>[</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></mrow><mrow><mi>N</mi></mrow></msup></mrow></math></span>. When the function of interest is the solution of a certain type of parametric PDE, recent results (Bonito et al., 2021 <span>[4]</span>) prove an upper bound on the number of spaces required to achieve a desired target accuracy. In this work, we prove a similar result for a more general class of functions with anisotropic analyticity, namely the class introduced in Bonito et al. (2021) <span>[5]</span>. In this way we show both where the theory developed in Bonito et al. (2021) <span>[4]</span> depends on being in the setting of parametric PDEs with affine diffusion coefficients, and prove a more general result outside of this setting.</p></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0021904524000261/pdfft?md5=37c84e9e2faa6a5470ebb676b660de8f&pid=1-s2.0-S0021904524000261-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140321216","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":"Wavelet characterization of exponentially weighted Besov space with dominating mixed smoothness and its application to function approximation","authors":"Yoshihiro Kogure,&nbsp;Ken’ichiro Tanaka","doi":"10.1016/j.jat.2024.106037","DOIUrl":"10.1016/j.jat.2024.106037","url":null,"abstract":"<div><p>Although numerous studies have focused on normal Besov spaces, limited studies have been conducted on exponentially weighted Besov spaces. Therefore, we define exponentially weighted Besov space <span><math><mrow><mi>V</mi><msubsup><mrow><mi>B</mi></mrow><mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow><mrow><mi>δ</mi><mo>,</mo><mi>w</mi></mrow></msubsup><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span> whose smoothness includes normal Besov spaces, Besov spaces with dominating mixed smoothness, and their interpolation. Furthermore, we obtain wavelet characterization of <span><math><mrow><mi>V</mi><msubsup><mrow><mi>B</mi></mrow><mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow><mrow><mi>δ</mi><mo>,</mo><mi>w</mi></mrow></msubsup><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span>. Next, approximation formulas such as sparse grids are derived using the determined formula. The results of this study are expected to provide considerable insight into the application of exponentially weighted Besov spaces with mixed smoothness.</p></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140127543","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}