Pub Date : 2025-06-01Epub Date: 2025-02-06DOI: 10.1016/j.jat.2025.106146
Magomed-Kasumov M.G
It is shown that for any absolutely continuous function on , the Fourier series with respect to the Jacobi polynomials converges uniformly on to this function if and only if , .
{"title":"Uniform convergence of Fourier–Jacobi series to absolutely continuous functions","authors":"Magomed-Kasumov M.G","doi":"10.1016/j.jat.2025.106146","DOIUrl":"10.1016/j.jat.2025.106146","url":null,"abstract":"<div><div>It is shown that for any absolutely continuous function on <span><math><mrow><mo>[</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></math></span>, the Fourier series with respect to the Jacobi polynomials <span><math><msubsup><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mo>,</mo><mi>β</mi></mrow></msubsup></math></span> converges uniformly on <span><math><mrow><mo>[</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></math></span> to this function if and only if <span><math><mrow><mo>−</mo><mn>1</mn><mo><</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>≤</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></math></span>, <span><math><mrow><mrow><mo>|</mo><mi>α</mi><mo>−</mo><mi>β</mi><mo>|</mo></mrow><mo>≤</mo><mn>1</mn></mrow></math></span>.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"308 ","pages":"Article 106146"},"PeriodicalIF":0.9,"publicationDate":"2025-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143379077","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}
Pub Date : 2025-06-01Epub Date: 2025-02-06DOI: 10.1016/j.jat.2025.106147
Ch. Boukeffous , A. San Antolín
The notion of simultaneous approximation order from shift-invariant subspaces in Sobolev spaces was introduced in the paper by Zhao (1995). Moreover, a characterization of those principal shift-invariant subspaces that provide simultaneous approximation order was proved there. In this note, we prove another characterization using dilated by some adequate expansive linear maps of a shift-invariant subspace. In addition, we introduce the notion of simultaneous density order and give necessary and sufficient conditions on a shift-invariant subspace to have a simultaneous density desired. To give our conditions, we shall explain the behavior on a neighborhood of the origin of the Fourier transform of the generators of a shift-invariant subspace. For this, we will use the classical notion of approximate continuity.
{"title":"On simultaneous density order from shift invariant subspaces in Sobolev spaces","authors":"Ch. Boukeffous , A. San Antolín","doi":"10.1016/j.jat.2025.106147","DOIUrl":"10.1016/j.jat.2025.106147","url":null,"abstract":"<div><div>The notion of simultaneous approximation order <span><math><mrow><mo>(</mo><mi>m</mi><mo>,</mo><mi>k</mi><mo>)</mo></mrow></math></span> from shift-invariant subspaces in Sobolev spaces was introduced in the paper by Zhao (1995). Moreover, a characterization of those principal shift-invariant subspaces that provide simultaneous approximation order <span><math><mrow><mo>(</mo><mi>m</mi><mo>,</mo><mi>k</mi><mo>)</mo></mrow></math></span> was proved there. In this note, we prove another characterization using dilated by some adequate expansive linear maps of a shift-invariant subspace. In addition, we introduce the notion of simultaneous density order <span><math><mrow><mo>(</mo><mi>m</mi><mo>,</mo><mi>k</mi><mo>)</mo></mrow></math></span> and give necessary and sufficient conditions on a shift-invariant subspace to have a simultaneous density desired. To give our conditions, we shall explain the behavior on a neighborhood of the origin of the Fourier transform of the generators of a shift-invariant subspace. For this, we will use the classical notion of approximate continuity.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"308 ","pages":"Article 106147"},"PeriodicalIF":0.9,"publicationDate":"2025-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143379119","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}
Pub Date : 2025-06-01Epub Date: 2025-02-11DOI: 10.1016/j.jat.2025.106149
Jacek Gulgowski , Anna Kamont , Markus Passenbrunner
<div><div>Let <span><math><mrow><mo>(</mo><mi>Ω</mi><mo>,</mo><mi>ℱ</mi><mo>,</mo><mi>P</mi><mo>)</mo></mrow></math></span> be a probability space and let <span><math><msubsup><mrow><mrow><mo>(</mo><msub><mrow><mi>ℱ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow></mrow><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>∞</mi></mrow></msubsup></math></span> be a binary filtration. i.e. exactly one atom of <span><math><msub><mrow><mi>ℱ</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></math></span> is divided into <em>two</em> atoms of <span><math><msub><mrow><mi>ℱ</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> without any restriction on their respective measures. Additionally, denote the collection of atoms corresponding to this filtration by <span><math><mi>A</mi></math></span>. Let <span><math><mrow><mi>S</mi><mo>⊂</mo><msup><mrow><mi>L</mi></mrow><mrow><mi>∞</mi></mrow></msup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow></mrow></math></span> be a finite-dimensional linear subspace, having an additional stability property on atoms <span><math><mi>A</mi></math></span>. For these data, we consider two dictionaries: <ul><li><span>•</span><span><div><span><math><mrow><mi>C</mi><mo>=</mo><mrow><mo>{</mo><mi>f</mi><mi>⋅</mi><msub><mrow><mi>1</mi></mrow><mrow><mi>A</mi></mrow></msub><mo>:</mo><mi>f</mi><mo>∈</mo><mi>S</mi><mo>,</mo><mi>A</mi><mo>∈</mo><mi>A</mi><mo>}</mo></mrow></mrow></math></span>,</div></span></li><li><span>•</span><span><div><span><math><mi>Φ</mi></math></span> – a local orthonormal system generated by <span><math><mi>S</mi></math></span> and the filtration <span><math><msubsup><mrow><mrow><mo>(</mo><msub><mrow><mi>ℱ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow></mrow><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>∞</mi></mrow></msubsup></math></span>.</div></span></li></ul></div><div>Let <span><math><mrow><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mrow><mo>(</mo><mi>S</mi><mo>)</mo></mrow><mo>=</mo><msub><mrow><mover><mrow><mi>span</mi></mrow><mo>¯</mo></mover></mrow><mrow><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow></mrow></msub><mi>C</mi><mo>=</mo><msub><mrow><mover><mrow><mi>span</mi></mrow><mo>¯</mo></mover></mrow><mrow><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow></mrow></msub><mi>Φ</mi></mrow></math></span>, with <span><math><mrow><mn>1</mn><mo><</mo><mi>p</mi><mo><</mo><mi>∞</mi></mrow></math></span>. We are interested in approximation spaces <span><math><mrow><msubsup><mrow><mi>A</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>α</mi></mrow></msubsup><mrow><mo>(</mo><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mrow><mo>(</mo><mi>S</mi><mo>)</mo></mrow><mo>,</mo><mi>C</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><msubsup><mrow><mi>A</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>α</mi></mrow></msubsup><mrow><mo>
设（Ω，，P）是一个概率空间，设(n)n=0∞是一个二元滤波。即，精确地将一个原子（n−1）分成两个原子（n−1）而不限制它们各自的度量。另外，用a表示与此过滤对应的原子集合。设S∧L∞（Ω）是一个有限维线性子空间，在原子a上具有额外的稳定性。对于这些数据，我们考虑两个字典：•C={f⋅1A:f∈S, a∈a}，•Φ—一个由S生成的局部正交系统和过滤(n)n=0∞。让Lp (S) =¯Lp(Ω)C =跨¯Lp(Ω)Φ1 & lt;术中;∞。我们对近似空间Aqα（Lp(S)，C）和Aqα（Lp(S)，Φ）感兴趣，它们分别对应于C和Φ的元素在Lp(S)中的最佳n项近似，其中α>；0和0<；q≤∞。已知在经典Haar情况下，即当S=span（1[0,1]）且二元滤除(n)n=0∞是二进的（即原子A∈A被分成两个新的等测度原子），我们有Aqα（Lp(S)，Φ）=Aqα(Lp(S)，C), cf. P. Petrushev（2003）。这促使我们问这样一个问题：在上述的一般情况下，这个等式是否成立？这个问题的答案取决于一个特定的Bernstein型不等式BI（a，S,p,τ）的有效性，其参数为1<；p<∞,0<τ<p。本文的主要结果是该类Bernstein不等式BI（a，S,p,τ）的一个几何刻划，即关于原子a和环上空间S上的函数的行为的一个刻划={a∈B: a,B∈a,B∧a}} a。我们将这个一般结果专门用于一些感兴趣的例子，包括一般Haar系统和由（多元）多项式组成的空间S。
{"title":"Properties of local orthonormal systems, Part II: Geometric characterization of Bernstein inequalities","authors":"Jacek Gulgowski , Anna Kamont , Markus Passenbrunner","doi":"10.1016/j.jat.2025.106149","DOIUrl":"10.1016/j.jat.2025.106149","url":null,"abstract":"<div><div>Let <span><math><mrow><mo>(</mo><mi>Ω</mi><mo>,</mo><mi>ℱ</mi><mo>,</mo><mi>P</mi><mo>)</mo></mrow></math></span> be a probability space and let <span><math><msubsup><mrow><mrow><mo>(</mo><msub><mrow><mi>ℱ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow></mrow><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>∞</mi></mrow></msubsup></math></span> be a binary filtration. i.e. exactly one atom of <span><math><msub><mrow><mi>ℱ</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></math></span> is divided into <em>two</em> atoms of <span><math><msub><mrow><mi>ℱ</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> without any restriction on their respective measures. Additionally, denote the collection of atoms corresponding to this filtration by <span><math><mi>A</mi></math></span>. Let <span><math><mrow><mi>S</mi><mo>⊂</mo><msup><mrow><mi>L</mi></mrow><mrow><mi>∞</mi></mrow></msup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow></mrow></math></span> be a finite-dimensional linear subspace, having an additional stability property on atoms <span><math><mi>A</mi></math></span>. For these data, we consider two dictionaries: <ul><li><span>•</span><span><div><span><math><mrow><mi>C</mi><mo>=</mo><mrow><mo>{</mo><mi>f</mi><mi>⋅</mi><msub><mrow><mi>1</mi></mrow><mrow><mi>A</mi></mrow></msub><mo>:</mo><mi>f</mi><mo>∈</mo><mi>S</mi><mo>,</mo><mi>A</mi><mo>∈</mo><mi>A</mi><mo>}</mo></mrow></mrow></math></span>,</div></span></li><li><span>•</span><span><div><span><math><mi>Φ</mi></math></span> – a local orthonormal system generated by <span><math><mi>S</mi></math></span> and the filtration <span><math><msubsup><mrow><mrow><mo>(</mo><msub><mrow><mi>ℱ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow></mrow><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>∞</mi></mrow></msubsup></math></span>.</div></span></li></ul></div><div>Let <span><math><mrow><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mrow><mo>(</mo><mi>S</mi><mo>)</mo></mrow><mo>=</mo><msub><mrow><mover><mrow><mi>span</mi></mrow><mo>¯</mo></mover></mrow><mrow><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow></mrow></msub><mi>C</mi><mo>=</mo><msub><mrow><mover><mrow><mi>span</mi></mrow><mo>¯</mo></mover></mrow><mrow><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow></mrow></msub><mi>Φ</mi></mrow></math></span>, with <span><math><mrow><mn>1</mn><mo><</mo><mi>p</mi><mo><</mo><mi>∞</mi></mrow></math></span>. We are interested in approximation spaces <span><math><mrow><msubsup><mrow><mi>A</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>α</mi></mrow></msubsup><mrow><mo>(</mo><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mrow><mo>(</mo><mi>S</mi><mo>)</mo></mrow><mo>,</mo><mi>C</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><msubsup><mrow><mi>A</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>α</mi></mrow></msubsup><mrow><mo>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"308 ","pages":"Article 106149"},"PeriodicalIF":0.9,"publicationDate":"2025-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143428911","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}
Pub Date : 2025-06-01Epub Date: 2025-02-13DOI: 10.1016/j.jat.2025.106151
Deimer J.J. Aleans , Sergio A. Tozoni
<div><div>In this paper we investigate the asymptotic behavior of entropy numbers of multiplier operators <span><math><msub><mrow><mi>Λ</mi></mrow><mrow><mo>∗</mo></mrow></msub></math></span> and <span><math><mi>Λ</mi></math></span>, defined for functions on the complex sphere <span><math><msub><mrow><mi>Ω</mi></mrow><mrow><mi>d</mi></mrow></msub></math></span> of <span><math><msup><mrow><mi>ℂ</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>, associated with sequences of multipliers of the type <span><math><msub><mrow><mrow><mo>{</mo><msubsup><mrow><mi>λ</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>n</mi></mrow><mrow><mo>∗</mo></mrow></msubsup><mo>}</mo></mrow></mrow><mrow><mi>m</mi><mo>,</mo><mi>n</mi><mo>∈</mo><mi>N</mi></mrow></msub></math></span>, <span><math><mrow><msubsup><mrow><mi>λ</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>n</mi></mrow><mrow><mo>∗</mo></mrow></msubsup><mo>=</mo><mi>λ</mi><mrow><mo>(</mo><mi>m</mi><mo>+</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><msub><mrow><mrow><mo>{</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>n</mi></mrow></msub><mo>}</mo></mrow></mrow><mrow><mi>m</mi><mo>,</mo><mi>n</mi><mo>∈</mo><mi>N</mi></mrow></msub></math></span>, <span><math><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>n</mi></mrow></msub><mo>=</mo><mi>λ</mi><mrow><mo>(</mo><mo>max</mo><mrow><mo>{</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>}</mo></mrow><mo>)</mo></mrow></mrow></math></span>, respectively, for a bounded function <span><math><mi>λ</mi></math></span> defined on <span><math><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow></math></span>. If the operators <span><math><msub><mrow><mi>Λ</mi></mrow><mrow><mo>∗</mo></mrow></msub></math></span> and <span><math><mi>Λ</mi></math></span> are bounded from <span><math><mrow><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mrow><mo>(</mo><msub><mrow><mi>Ω</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span> into <span><math><mrow><msup><mrow><mi>L</mi></mrow><mrow><mi>q</mi></mrow></msup><mrow><mo>(</mo><msub><mrow><mi>Ω</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span>, <span><math><mrow><mn>1</mn><mo>≤</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>≤</mo><mi>∞</mi></mrow></math></span>, and <span><math><msub><mrow><mi>U</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> is the closed unit ball of <span><math><mrow><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mrow><mo>(</mo><msub><mrow><mi>Ω</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span>, we study lower and upper estimates for the entropy numbers of the sets <span><math><mrow><msub><mrow><mi>Λ</mi></mrow><mrow><mo>∗</mo></mrow></msub><msub><mrow><mi>U</mi></mrow><mrow><mi>p</mi></mrow></msub></mrow></math></span> and <span><math><mrow><mi>Λ</mi><msub><mrow><mi>U</mi></mrow><mrow><mi>p</mi></mrow></msub></mrow></math></span> in <span><math><mrow><msup><mrow><mi
{"title":"Estimates for entropy numbers of sets of smooth functions on complex spheres","authors":"Deimer J.J. Aleans , Sergio A. Tozoni","doi":"10.1016/j.jat.2025.106151","DOIUrl":"10.1016/j.jat.2025.106151","url":null,"abstract":"<div><div>In this paper we investigate the asymptotic behavior of entropy numbers of multiplier operators <span><math><msub><mrow><mi>Λ</mi></mrow><mrow><mo>∗</mo></mrow></msub></math></span> and <span><math><mi>Λ</mi></math></span>, defined for functions on the complex sphere <span><math><msub><mrow><mi>Ω</mi></mrow><mrow><mi>d</mi></mrow></msub></math></span> of <span><math><msup><mrow><mi>ℂ</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>, associated with sequences of multipliers of the type <span><math><msub><mrow><mrow><mo>{</mo><msubsup><mrow><mi>λ</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>n</mi></mrow><mrow><mo>∗</mo></mrow></msubsup><mo>}</mo></mrow></mrow><mrow><mi>m</mi><mo>,</mo><mi>n</mi><mo>∈</mo><mi>N</mi></mrow></msub></math></span>, <span><math><mrow><msubsup><mrow><mi>λ</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>n</mi></mrow><mrow><mo>∗</mo></mrow></msubsup><mo>=</mo><mi>λ</mi><mrow><mo>(</mo><mi>m</mi><mo>+</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><msub><mrow><mrow><mo>{</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>n</mi></mrow></msub><mo>}</mo></mrow></mrow><mrow><mi>m</mi><mo>,</mo><mi>n</mi><mo>∈</mo><mi>N</mi></mrow></msub></math></span>, <span><math><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>n</mi></mrow></msub><mo>=</mo><mi>λ</mi><mrow><mo>(</mo><mo>max</mo><mrow><mo>{</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>}</mo></mrow><mo>)</mo></mrow></mrow></math></span>, respectively, for a bounded function <span><math><mi>λ</mi></math></span> defined on <span><math><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow></math></span>. If the operators <span><math><msub><mrow><mi>Λ</mi></mrow><mrow><mo>∗</mo></mrow></msub></math></span> and <span><math><mi>Λ</mi></math></span> are bounded from <span><math><mrow><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mrow><mo>(</mo><msub><mrow><mi>Ω</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span> into <span><math><mrow><msup><mrow><mi>L</mi></mrow><mrow><mi>q</mi></mrow></msup><mrow><mo>(</mo><msub><mrow><mi>Ω</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span>, <span><math><mrow><mn>1</mn><mo>≤</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>≤</mo><mi>∞</mi></mrow></math></span>, and <span><math><msub><mrow><mi>U</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> is the closed unit ball of <span><math><mrow><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mrow><mo>(</mo><msub><mrow><mi>Ω</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span>, we study lower and upper estimates for the entropy numbers of the sets <span><math><mrow><msub><mrow><mi>Λ</mi></mrow><mrow><mo>∗</mo></mrow></msub><msub><mrow><mi>U</mi></mrow><mrow><mi>p</mi></mrow></msub></mrow></math></span> and <span><math><mrow><mi>Λ</mi><msub><mrow><mi>U</mi></mrow><mrow><mi>p</mi></mrow></msub></mrow></math></span> in <span><math><mrow><msup><mrow><mi","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"308 ","pages":"Article 106151"},"PeriodicalIF":0.9,"publicationDate":"2025-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143420993","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}
Pub Date : 2025-05-01Epub Date: 2024-12-21DOI: 10.1016/j.jat.2024.106142
A.S. Jooste , K. Jordaan
In this paper we consider connection formulae for orthogonal polynomials in the context of Christoffel transformations for the case where a weight function, not necessarily even, is multiplied by an even function , to determine new lower bounds for the largest zero and upper bounds for the smallest zero of a Meixner–Pollaczek polynomial. When is orthogonal with respect to a weight and is orthogonal with respect to the weight , we show that is a necessary and sufficient condition for existence of a connection formula involving a polynomial of degree , such that the zeros of and the zeros of interlace. We analyse the new inner bounds for the extreme zeros of Meixner–Pollaczek polynomials to determine which bounds are the sharpest. We also briefly discuss bounds for the zeros of Pseudo-Jacobi polynomials.
{"title":"Bounds for extreme zeros of Meixner–Pollaczek polynomials","authors":"A.S. Jooste , K. Jordaan","doi":"10.1016/j.jat.2024.106142","DOIUrl":"10.1016/j.jat.2024.106142","url":null,"abstract":"<div><div>In this paper we consider connection formulae for orthogonal polynomials in the context of Christoffel transformations for the case where a weight function, not necessarily even, is multiplied by an even function <span><math><mrow><msub><mrow><mi>c</mi></mrow><mrow><mn>2</mn><mi>k</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>,</mo><mi>k</mi><mo>∈</mo><msub><mrow><mi>N</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></math></span>, to determine new lower bounds for the largest zero and upper bounds for the smallest zero of a Meixner–Pollaczek polynomial. When <span><math><msub><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is orthogonal with respect to a weight <span><math><mrow><mi>w</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><msub><mrow><mi>g</mi></mrow><mrow><mi>n</mi><mo>−</mo><mi>m</mi></mrow></msub></math></span> is orthogonal with respect to the weight <span><math><mrow><msub><mrow><mi>c</mi></mrow><mrow><mn>2</mn><mi>k</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mi>w</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span>, we show that <span><math><mrow><mi>k</mi><mo>∈</mo><mrow><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>m</mi><mo>}</mo></mrow></mrow></math></span> is a necessary and sufficient condition for existence of a connection formula involving a polynomial <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msub></math></span> of degree <span><math><mrow><mo>(</mo><mi>m</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></math></span>, such that the <span><math><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></math></span> zeros of <span><math><mrow><msub><mrow><mi>G</mi></mrow><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msub><msub><mrow><mi>g</mi></mrow><mrow><mi>n</mi><mo>−</mo><mi>m</mi></mrow></msub></mrow></math></span> and the <span><math><mi>n</mi></math></span> zeros of <span><math><msub><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> interlace. We analyse the new inner bounds for the extreme zeros of Meixner–Pollaczek polynomials to determine which bounds are the sharpest. We also briefly discuss bounds for the zeros of Pseudo-Jacobi polynomials.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"307 ","pages":"Article 106142"},"PeriodicalIF":0.9,"publicationDate":"2025-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143154842","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}
Pub Date : 2025-05-01Epub Date: 2025-01-13DOI: 10.1016/j.jat.2025.106143
Tanay Wakhare
We embark on a systematic study of the -th derivative of , where is the binary entropy and are integers. Our motivation is the conjectural entropy inequality , where is given by a functional equation. The case was the key technical tool driving recent breakthroughs on the union-closed sets conjecture. We express as a rational function, an infinite series, and a sum over generalized Stirling numbers. This allows us to reduce the proof of the entropy inequality for real to showing that an associated polynomial has only two real roots in the interval , which also allows us to prove the inequality for fractional exponents such as . The proof suggests a new framework for proving tight inequalities for the sum of polynomials times the logarithms of polynomials, which converts the inequality into a statement about the real roots of a simpler associated polynomial.
{"title":"Iterated entropy derivatives and binary entropy inequalities","authors":"Tanay Wakhare","doi":"10.1016/j.jat.2025.106143","DOIUrl":"10.1016/j.jat.2025.106143","url":null,"abstract":"<div><div>We embark on a systematic study of the <span><math><mrow><mo>(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></math></span>-th derivative of <span><math><mrow><msup><mrow><mi>x</mi></mrow><mrow><mi>k</mi><mo>−</mo><mi>r</mi></mrow></msup><mi>H</mi><mrow><mo>(</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>r</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span>, where <span><math><mrow><mi>H</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>≔</mo><mo>−</mo><mi>x</mi><mo>log</mo><mi>x</mi><mo>−</mo><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>x</mi><mo>)</mo></mrow><mo>log</mo><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span> is the binary entropy and <span><math><mrow><mi>k</mi><mo>≥</mo><mi>r</mi><mo>≥</mo><mn>1</mn></mrow></math></span> are integers. Our motivation is the conjectural entropy inequality <span><math><mrow><msub><mrow><mi>α</mi></mrow><mrow><mi>k</mi></mrow></msub><mi>H</mi><mrow><mo>(</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>)</mo></mrow><mo>≥</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>H</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span>, where <span><math><mrow><mn>0</mn><mo><</mo><msub><mrow><mi>α</mi></mrow><mrow><mi>k</mi></mrow></msub><mo><</mo><mn>1</mn></mrow></math></span> is given by a functional equation. The <span><math><mrow><mi>k</mi><mo>=</mo><mn>2</mn></mrow></math></span> case was the key technical tool driving recent breakthroughs on the union-closed sets conjecture. We express <span><math><mrow><mfrac><mrow><msup><mrow><mi>d</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><mrow><mi>d</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow></mfrac><msup><mrow><mi>x</mi></mrow><mrow><mi>k</mi><mo>−</mo><mi>r</mi></mrow></msup><mi>H</mi><mrow><mo>(</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>r</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span> as a rational function, an infinite series, and a sum over generalized Stirling numbers. This allows us to reduce the proof of the entropy inequality for real <span><math><mi>k</mi></math></span> to showing that an associated polynomial has only two real roots in the interval <span><math><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></math></span>, which also allows us to prove the inequality for fractional exponents such as <span><math><mrow><mi>k</mi><mo>=</mo><mn>3</mn><mo>/</mo><mn>2</mn></mrow></math></span>. The proof suggests a new framework for proving tight inequalities for the sum of polynomials times the logarithms of polynomials, which converts the inequality into a statement about the real roots of a simpler associated polynomial.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"307 ","pages":"Article 106143"},"PeriodicalIF":0.9,"publicationDate":"2025-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143154843","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}
Pub Date : 2025-05-01Epub Date: 2025-01-11DOI: 10.1016/j.jat.2025.106144
Tommaso Bruno , Federico Santagati
We give optimal bounds for the radial, space and time derivatives of arbitrary order of the heat kernel of the Laplace–Beltrami operator on Damek–Ricci spaces. In the case of symmetric spaces of rank one, these complete and actually improve conjectured estimates by Anker and Ji. We also provide asymptotics at infinity of all the radial and time derivates of the kernel. Along the way, we provide sharp bounds for all the derivatives of the Riemannian distance and obtain analogous bounds for those of the heat kernel of the distinguished Laplacian.
{"title":"Optimal heat kernel bounds and asymptotics on Damek–Ricci spaces","authors":"Tommaso Bruno , Federico Santagati","doi":"10.1016/j.jat.2025.106144","DOIUrl":"10.1016/j.jat.2025.106144","url":null,"abstract":"<div><div>We give optimal bounds for the radial, space and time derivatives of arbitrary order of the heat kernel of the Laplace–Beltrami operator on Damek–Ricci spaces. In the case of symmetric spaces of rank one, these complete and actually improve conjectured estimates by Anker and Ji. We also provide asymptotics at infinity of all the radial and time derivates of the kernel. Along the way, we provide sharp bounds for all the derivatives of the Riemannian distance and obtain analogous bounds for those of the heat kernel of the distinguished Laplacian.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"307 ","pages":"Article 106144"},"PeriodicalIF":0.9,"publicationDate":"2025-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143154844","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}
Pub Date : 2025-05-01Epub Date: 2024-12-21DOI: 10.1016/j.jat.2024.106141
Fernando Albiac , José L. Ansorena , Vladimir Temlyakov
Although the basic idea behind the concept of a greedy basis had been around for some time, the formal development of a theory of greedy bases was initiated in 1999 with the publication of the article [S.V. Konyagin and V.N. Temlyakov, A remark on greedy approximation in Banach spaces, East J. Approx. 5 (3) (1999), 365-379]. The theoretical simplicity of the thresholding greedy algorithm became a model for a procedure widely used in numerical applications and the subject of greedy bases evolved very rapidly from the point of view of approximation theory. The idea of studying greedy bases and related greedy algorithms attracted also the attention of researchers with a classical Banach space theory background. From the more abstract point of functional analysis, the theory of greedy bases and its derivates evolved very fast as many fundamental results were discovered and new ramifications branched out. Hundreds of papers on greedy-like bases and several monographs have been written since the appearance of the aforementioned foundational paper. After twenty-five years, the theory is very much alive and it continues to be a very active research topic both for functional analysts and for researchers interested in the applied nature of nonlinear approximation alike. This is why we believe it is a good moment to gather a selection of 25 open problems (one per year since 1999!) whose solution would contribute to advance the state of art of this beautiful topic.
{"title":"Twenty-five years of greedy bases","authors":"Fernando Albiac , José L. Ansorena , Vladimir Temlyakov","doi":"10.1016/j.jat.2024.106141","DOIUrl":"10.1016/j.jat.2024.106141","url":null,"abstract":"<div><div>Although the basic idea behind the concept of a greedy basis had been around for some time, the formal development of a theory of greedy bases was initiated in 1999 with the publication of the article [S.V. Konyagin and V.N. Temlyakov, A remark on greedy approximation in Banach spaces, East J. Approx. 5 (3) (1999), 365-379]. The theoretical simplicity of the thresholding greedy algorithm became a model for a procedure widely used in numerical applications and the subject of greedy bases evolved very rapidly from the point of view of approximation theory. The idea of studying greedy bases and related greedy algorithms attracted also the attention of researchers with a classical Banach space theory background. From the more abstract point of functional analysis, the theory of greedy bases and its derivates evolved very fast as many fundamental results were discovered and new ramifications branched out. Hundreds of papers on greedy-like bases and several monographs have been written since the appearance of the aforementioned foundational paper. After twenty-five years, the theory is very much alive and it continues to be a very active research topic both for functional analysts and for researchers interested in the applied nature of nonlinear approximation alike. This is why we believe it is a good moment to gather a selection of 25 open problems (one per year since 1999!) whose solution would contribute to advance the state of art of this beautiful topic.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"307 ","pages":"Article 106141"},"PeriodicalIF":0.9,"publicationDate":"2025-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143154841","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}
Pub Date : 2025-03-01Epub Date: 2024-11-22DOI: 10.1016/j.jat.2024.106126
Erlan D. Nursultanov , Durvudkhan Suragan
In this paper, we present a refined version of the (classical) Stein inequality for the Fourier transform, elevating it to a new level of accuracy. Furthermore, we establish extended analogues of a more precise version of the Stein inequality for the Fourier transform, broadening its applicability from the range to .
{"title":"Improved Stein inequalities for the Fourier transform","authors":"Erlan D. Nursultanov , Durvudkhan Suragan","doi":"10.1016/j.jat.2024.106126","DOIUrl":"10.1016/j.jat.2024.106126","url":null,"abstract":"<div><div>In this paper, we present a refined version of the (classical) Stein inequality for the Fourier transform, elevating it to a new level of accuracy. Furthermore, we establish extended analogues of a more precise version of the Stein inequality for the Fourier transform, broadening its applicability from the range <span><math><mrow><mn>1</mn><mo><</mo><mi>p</mi><mo><</mo><mn>2</mn></mrow></math></span> to <span><math><mrow><mn>2</mn><mo>≤</mo><mi>p</mi><mo><</mo><mi>∞</mi></mrow></math></span>.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"306 ","pages":"Article 106126"},"PeriodicalIF":0.9,"publicationDate":"2025-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142722538","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}
Pub Date : 2025-03-01Epub Date: 2024-11-15DOI: 10.1016/j.jat.2024.106122
Tamás Erdélyi
We prove that there is an absolute constant such that for every there are such that the polynomial of the form has at least distinct sign changes in , where . This improves and extends earlier results of Bloch and Pólya and Erdélyi and, as a special case, recaptures a special case of a more general recent result of Jacob and Nazarov.
我们证明存在一个绝对常数 c>0，使得对于每一个 a0,a1,...,an∈[1,M],1≤M≤14expn9,有 b0,b1,...,bn∈{-1,0,1}，使得形式为 P(z)=∑j=0nbjajzj 的多项式 P 在 Ia 中至少有 cnlog(4M)1/2-1 个不同的符号变化：=(1-2a,1-a)，其中 a:=log(4M)n1/2≤1/3.这改进并扩展了布洛赫、波利亚和埃尔德利的早期结果，并作为一个特例，重现了雅各布和纳扎罗夫的一个更普遍的最新结果的特例。
{"title":"Extensions of the Bloch-Pólya theorem on the number of real zeros of polynomials (II)","authors":"Tamás Erdélyi","doi":"10.1016/j.jat.2024.106122","DOIUrl":"10.1016/j.jat.2024.106122","url":null,"abstract":"<div><div>We prove that there is an absolute constant <span><math><mrow><mi>c</mi><mo>></mo><mn>0</mn></mrow></math></span> such that for every <span><span><span><math><mrow><msub><mrow><mi>a</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>∈</mo><mrow><mo>[</mo><mn>1</mn><mo>,</mo><mi>M</mi><mo>]</mo></mrow><mspace></mspace><mo>,</mo><mspace></mspace><mn>1</mn><mo>≤</mo><mi>M</mi><mo>≤</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mo>exp</mo><mfenced><mrow><mfrac><mrow><mi>n</mi></mrow><mrow><mn>9</mn></mrow></mfrac></mrow></mfenced><mo>,</mo></mrow></math></span></span></span>there are <span><span><span><math><mrow><msub><mrow><mi>b</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>b</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>b</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>∈</mo><mrow><mo>{</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow></mrow></math></span></span></span>such that the polynomial <span><math><mi>P</mi></math></span> of the form <span><math><mrow><mi>P</mi><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>=</mo><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>j</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>n</mi></mrow></msubsup><msub><mrow><mi>b</mi></mrow><mrow><mi>j</mi></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mi>j</mi></mrow></msub><msup><mrow><mi>z</mi></mrow><mrow><mi>j</mi></mrow></msup></mrow></math></span> has at least <span><math><mrow><mi>c</mi><msup><mrow><mfenced><mrow><mfrac><mrow><mi>n</mi></mrow><mrow><mo>log</mo><mrow><mo>(</mo><mn>4</mn><mi>M</mi><mo>)</mo></mrow></mrow></mfrac></mrow></mfenced></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>−</mo><mn>1</mn></mrow></math></span> distinct sign changes in <span><math><mrow><msub><mrow><mi>I</mi></mrow><mrow><mi>a</mi></mrow></msub><mo>:</mo><mo>=</mo><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mn>2</mn><mi>a</mi><mo>,</mo><mn>1</mn><mo>−</mo><mi>a</mi><mo>)</mo></mrow></mrow></math></span>, where <span><math><mrow><mi>a</mi><mo>:</mo><mo>=</mo><msup><mrow><mfenced><mrow><mfrac><mrow><mo>log</mo><mrow><mo>(</mo><mn>4</mn><mi>M</mi><mo>)</mo></mrow></mrow><mrow><mi>n</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>≤</mo><mn>1</mn><mo>/</mo><mn>3</mn></mrow></math></span>. This improves and extends earlier results of Bloch and Pólya and Erdélyi and, as a special case, recaptures a special case of a more general recent result of Jacob and Nazarov.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"306 ","pages":"Article 106122"},"PeriodicalIF":0.9,"publicationDate":"2025-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142706248","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}
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