{"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":"2025-02-13","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}
Jacek Gulgowski , Anna Kamont , Markus Passenbrunner
{"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":"2025-02-11","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}
Carlos Beltrán , Damir Ferizović , Pedro R. López-Gómez
{"title":"Measure-preserving mappings from the unit cube to some symmetric spaces","authors":"Carlos Beltrán , Damir Ferizović , Pedro R. López-Gómez","doi":"10.1016/j.jat.2025.106145","DOIUrl":"10.1016/j.jat.2025.106145","url":null,"abstract":"<div><div>We construct measure-preserving mappings from the <span><math><mi>d</mi></math></span>-dimensional unit cube to the <span><math><mi>d</mi></math></span>-dimensional unit ball and the compact rank one symmetric spaces, namely the <span><math><mi>d</mi></math></span>-dimensional sphere, the real, complex, and quaternionic projective spaces, and the Cayley plane. We also give a procedure to generate measure-preserving mappings from the <span><math><mi>d</mi></math></span>-dimensional unit cube to product spaces and fiber bundles under certain conditions.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"308 ","pages":"Article 106145"},"PeriodicalIF":0.9,"publicationDate":"2025-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143379078","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":"Scaling limits of complex and symplectic non-Hermitian Wishart ensembles","authors":"Sung-Soo Byun , Kohei Noda","doi":"10.1016/j.jat.2025.106148","DOIUrl":"10.1016/j.jat.2025.106148","url":null,"abstract":"<div><div>Non-Hermitian Wishart matrices were introduced in the context of quantum chromodynamics with a baryon chemical potential. These provide chiral extensions of the elliptic Ginibre ensembles as well as non-Hermitian extensions of the classical Wishart/Laguerre ensembles. In this work, we investigate eigenvalues of non-Hermitian Wishart matrices in the symmetry classes of complex and symplectic Ginibre ensembles. We introduce a generalised Christoffel–Darboux formula in the form of a certain second-order differential equation, offering a unified and robust method for analysing correlation functions across all scaling regimes in the model. By employing this method, we derive bulk and edge scaling limits for eigenvalue correlations at both strong and weak non-Hermiticity.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"308 ","pages":"Article 106148"},"PeriodicalIF":0.9,"publicationDate":"2025-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143348474","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":"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":"2025-02-06","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}
{"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":"2025-02-06","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}
{"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":"2025-01-13","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}
{"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":"2025-01-11","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}
{"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":"2024-12-21","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}
Fernando Albiac , José L. Ansorena , Vladimir Temlyakov
{"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":"2024-12-21","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}
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