Pub Date : 2025-08-20DOI: 10.1016/j.jat.2025.106230
Monojit Bhattacharjee , Rajkamal Nailwal
In this article, we characterize completely alternating functions on an abelian semigroup in terms of completely monotone functions on the product semigroup . We also discuss completely alternating sequences induced by a class of rational functions and obtain a set of sufficient conditions (in terms of its zeros and poles) to determine them. As an application, we show a complete characterization of several classes of completely monotone functions on induced by rational functions in two variables. We also derive a set of necessary conditions for the complete monotonicity of the sequence .
{"title":"A characterization of completely alternating functions","authors":"Monojit Bhattacharjee , Rajkamal Nailwal","doi":"10.1016/j.jat.2025.106230","DOIUrl":"10.1016/j.jat.2025.106230","url":null,"abstract":"<div><div>In this article, we characterize completely alternating functions on an abelian semigroup <span><math><mi>S</mi></math></span> in terms of completely monotone functions on the product semigroup <span><math><mrow><mi>S</mi><mo>×</mo><msub><mrow><mi>Z</mi></mrow><mrow><mo>+</mo></mrow></msub></mrow></math></span>. We also discuss completely alternating sequences induced by a class of rational functions and obtain a set of sufficient conditions (in terms of its zeros and poles) to determine them. As an application, we show a complete characterization of several classes of completely monotone functions on <span><math><msubsup><mrow><mi>Z</mi></mrow><mrow><mo>+</mo></mrow><mrow><mn>2</mn></mrow></msubsup></math></span> induced by rational functions in two variables. We also derive a set of necessary conditions for the complete monotonicity of the sequence <span><math><mrow><msub><mrow><mrow><mo>{</mo><msubsup><mrow><mo>∏</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>k</mi></mrow></msubsup><mfrac><mrow><mrow><mo>(</mo><mi>n</mi><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo></mrow></mrow><mrow><mrow><mo>(</mo><mi>n</mi><mo>+</mo><msub><mrow><mi>b</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo></mrow></mrow></mfrac><mo>}</mo></mrow></mrow><mrow><mi>n</mi><mo>∈</mo><msub><mrow><mi>Z</mi></mrow><mrow><mo>+</mo></mrow></msub></mrow></msub><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>b</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∈</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow></mrow></math></span>.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"313 ","pages":"Article 106230"},"PeriodicalIF":0.6,"publicationDate":"2025-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144890100","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 : 2025-08-19DOI: 10.1016/j.jat.2025.106221
Alexander Brudnyi
<div><div>Let <span><math><mrow><msup><mrow><mi>D</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>⊂</mo><msup><mrow><mi>ℂ</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></math></span> denote the open unit polydisk, and let <span><math><mrow><mi>K</mi><mo>⊂</mo><msup><mrow><mi>D</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></math></span> be a Cartesian product of planar compacta. Let <span><math><mrow><mover><mrow><mi>U</mi></mrow><mrow><mo>̂</mo></mrow></mover><mo>⊂</mo><msup><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></math></span> be an open neighborhood of the closure <span><math><mover><mrow><mi>K</mi></mrow><mrow><mo>̄</mo></mrow></mover></math></span> in <span><math><msup><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>, where <span><math><mi>M</mi></math></span> is the maximal ideal space of the algebra <span><math><msup><mrow><mi>H</mi></mrow><mrow><mi>∞</mi></mrow></msup></math></span> of bounded holomorphic functions on <span><math><mi>D</mi></math></span>. Given a complex Banach space <span><math><mi>X</mi></math></span>, denote by <span><math><mrow><msup><mrow><mi>H</mi></mrow><mrow><mi>∞</mi></mrow></msup><mrow><mo>(</mo><mi>V</mi><mo>,</mo><mi>X</mi><mo>)</mo></mrow></mrow></math></span> the Banach space of bounded <span><math><mi>X</mi></math></span>-valued holomorphic functions on an open set <span><math><mrow><mi>V</mi><mo>⊂</mo><msup><mrow><mi>D</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></math></span>.</div><div>We prove that any <span><math><mrow><mi>f</mi><mo>∈</mo><msup><mrow><mi>H</mi></mrow><mrow><mi>∞</mi></mrow></msup><mrow><mo>(</mo><mi>U</mi><mo>,</mo><mi>X</mi><mo>)</mo></mrow></mrow></math></span>, where <span><math><mrow><mi>U</mi><mo>=</mo><mover><mrow><mi>U</mi></mrow><mrow><mo>̂</mo></mrow></mover><mo>∩</mo><msup><mrow><mi>D</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></math></span>, can be uniformly approximated on <span><math><mi>K</mi></math></span> by functions of the form <span><math><mrow><mi>h</mi><mo>/</mo><mi>b</mi></mrow></math></span>, where <span><math><mrow><mi>h</mi><mo>∈</mo><msup><mrow><mi>H</mi></mrow><mrow><mi>∞</mi></mrow></msup><mrow><mo>(</mo><msup><mrow><mi>D</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>,</mo><mi>X</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mi>b</mi></math></span> is a finite product of interpolating Blaschke products satisfying <span><math><mrow><msub><mrow><mo>inf</mo></mrow><mrow><mi>K</mi></mrow></msub><mrow><mo>|</mo><mi>b</mi><mo>|</mo></mrow><mo>></mo><mn>0</mn></mrow></math></span>. Moreover, if <span><math><mover><mrow><mi>K</mi></mrow><mrow><mo>̄</mo></mrow></mover></math></span> is contained in a compact holomorphically convex subset of <span><math><mover><mrow><mi>U</mi></mrow><mrow><mo>̂</mo></mrow></mover></math></span>, then such approximations can be achieved without denominators: that is, <span><math><mi>f</mi></math></span> can be approximated uniformly on <span><math><mi>K</mi></math></span>
{"title":"Runge-type approximation theorem for Banach-valued H∞ functions on a polydisk","authors":"Alexander Brudnyi","doi":"10.1016/j.jat.2025.106221","DOIUrl":"10.1016/j.jat.2025.106221","url":null,"abstract":"<div><div>Let <span><math><mrow><msup><mrow><mi>D</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>⊂</mo><msup><mrow><mi>ℂ</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></math></span> denote the open unit polydisk, and let <span><math><mrow><mi>K</mi><mo>⊂</mo><msup><mrow><mi>D</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></math></span> be a Cartesian product of planar compacta. Let <span><math><mrow><mover><mrow><mi>U</mi></mrow><mrow><mo>̂</mo></mrow></mover><mo>⊂</mo><msup><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></math></span> be an open neighborhood of the closure <span><math><mover><mrow><mi>K</mi></mrow><mrow><mo>̄</mo></mrow></mover></math></span> in <span><math><msup><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>, where <span><math><mi>M</mi></math></span> is the maximal ideal space of the algebra <span><math><msup><mrow><mi>H</mi></mrow><mrow><mi>∞</mi></mrow></msup></math></span> of bounded holomorphic functions on <span><math><mi>D</mi></math></span>. Given a complex Banach space <span><math><mi>X</mi></math></span>, denote by <span><math><mrow><msup><mrow><mi>H</mi></mrow><mrow><mi>∞</mi></mrow></msup><mrow><mo>(</mo><mi>V</mi><mo>,</mo><mi>X</mi><mo>)</mo></mrow></mrow></math></span> the Banach space of bounded <span><math><mi>X</mi></math></span>-valued holomorphic functions on an open set <span><math><mrow><mi>V</mi><mo>⊂</mo><msup><mrow><mi>D</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></math></span>.</div><div>We prove that any <span><math><mrow><mi>f</mi><mo>∈</mo><msup><mrow><mi>H</mi></mrow><mrow><mi>∞</mi></mrow></msup><mrow><mo>(</mo><mi>U</mi><mo>,</mo><mi>X</mi><mo>)</mo></mrow></mrow></math></span>, where <span><math><mrow><mi>U</mi><mo>=</mo><mover><mrow><mi>U</mi></mrow><mrow><mo>̂</mo></mrow></mover><mo>∩</mo><msup><mrow><mi>D</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></math></span>, can be uniformly approximated on <span><math><mi>K</mi></math></span> by functions of the form <span><math><mrow><mi>h</mi><mo>/</mo><mi>b</mi></mrow></math></span>, where <span><math><mrow><mi>h</mi><mo>∈</mo><msup><mrow><mi>H</mi></mrow><mrow><mi>∞</mi></mrow></msup><mrow><mo>(</mo><msup><mrow><mi>D</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>,</mo><mi>X</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mi>b</mi></math></span> is a finite product of interpolating Blaschke products satisfying <span><math><mrow><msub><mrow><mo>inf</mo></mrow><mrow><mi>K</mi></mrow></msub><mrow><mo>|</mo><mi>b</mi><mo>|</mo></mrow><mo>></mo><mn>0</mn></mrow></math></span>. Moreover, if <span><math><mover><mrow><mi>K</mi></mrow><mrow><mo>̄</mo></mrow></mover></math></span> is contained in a compact holomorphically convex subset of <span><math><mover><mrow><mi>U</mi></mrow><mrow><mo>̂</mo></mrow></mover></math></span>, then such approximations can be achieved without denominators: that is, <span><math><mi>f</mi></math></span> can be approximated uniformly on <span><math><mi>K</mi></math></span>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"314 ","pages":"Article 106221"},"PeriodicalIF":0.6,"publicationDate":"2025-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145683674","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 : 2025-08-19DOI: 10.1016/j.jat.2025.106227
Gökalp Alpan , Turgay Bayraktar , Norm Levenberg
We generalize the theory of Widom factors to the setting. We define Widom factors of compact subsets associated with multivariate orthogonal polynomials and weighted Chebyshev polynomials. We show that on product subsets of , where each is a non-polar compact subset of , these quantities have universal lower bounds which directly extend one dimensional results. Under the additional assumption that each is a subset of the real line, we provide improved lower bounds for Widom factors for some weight functions ; in particular, for the case . Finally, we define the Mahler measure of a multivariate polynomial relative to and obtain lower bounds for this quantity on product sets.
{"title":"Widom factors in ℂn","authors":"Gökalp Alpan , Turgay Bayraktar , Norm Levenberg","doi":"10.1016/j.jat.2025.106227","DOIUrl":"10.1016/j.jat.2025.106227","url":null,"abstract":"<div><div>We generalize the theory of Widom factors to the <span><math><msup><mrow><mi>ℂ</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> setting. We define Widom factors of compact subsets <span><math><mrow><mi>K</mi><mo>⊂</mo><msup><mrow><mi>ℂ</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></math></span> associated with multivariate orthogonal polynomials and weighted Chebyshev polynomials. We show that on product subsets <span><math><mrow><mi>K</mi><mo>=</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>×</mo><mo>⋯</mo><mo>×</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></math></span> of <span><math><msup><mrow><mi>ℂ</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>, where each <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>j</mi></mrow></msub></math></span> is a non-polar compact subset of <span><math><mi>ℂ</mi></math></span>, these quantities have universal lower bounds which directly extend one dimensional results. Under the additional assumption that each <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>j</mi></mrow></msub></math></span> is a subset of the real line, we provide improved lower bounds for Widom factors for some weight functions <span><math><mi>w</mi></math></span>; in particular, for the case <span><math><mrow><mi>w</mi><mo>≡</mo><mn>1</mn></mrow></math></span>. Finally, we define the Mahler measure of a multivariate polynomial relative to <span><math><mrow><mi>K</mi><mo>⊂</mo><msup><mrow><mi>ℂ</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></math></span> and obtain lower bounds for this quantity on product sets.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"313 ","pages":"Article 106227"},"PeriodicalIF":0.6,"publicationDate":"2025-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144887235","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 : 2025-08-19DOI: 10.1016/j.jat.2025.106229
Kilian Hermann, Michael Voit
For the -Hermite, Laguerre, and Jacobi ensembles of dimension there exist central limit theorems for the freezing case such that the associated means and covariances can be expressed in terms of the associated Hermite, Laguerre, and Jacobi polynomials of order respectively as well as via the associated dual polynomials in the sense of de Boor and Saff. In this paper we derive limits for for the covariances of the largest (and smallest) eigenvalues for these frozen Jacobi ensembles in terms of Bessel functions. These results correspond to the hard edge analysis in the frozen Laguerre cases by Andraus and Lerner-Brecher and to known results for finite .
{"title":"On the extremal eigenvalues of Jacobi ensembles at zero temperature","authors":"Kilian Hermann, Michael Voit","doi":"10.1016/j.jat.2025.106229","DOIUrl":"10.1016/j.jat.2025.106229","url":null,"abstract":"<div><div>For the <span><math><mi>β</mi></math></span>-Hermite, Laguerre, and Jacobi ensembles of dimension <span><math><mi>N</mi></math></span> there exist central limit theorems for the freezing case <span><math><mrow><mi>β</mi><mo>→</mo><mi>∞</mi></mrow></math></span> such that the associated means and covariances can be expressed in terms of the associated Hermite, Laguerre, and Jacobi polynomials of order <span><math><mi>N</mi></math></span> respectively as well as via the associated dual polynomials in the sense of de Boor and Saff. In this paper we derive limits for <span><math><mrow><mi>N</mi><mo>→</mo><mi>∞</mi></mrow></math></span> for the covariances of the <span><math><mrow><mi>r</mi><mo>∈</mo><mi>N</mi></mrow></math></span> largest (and smallest) eigenvalues for these frozen Jacobi ensembles in terms of Bessel functions. These results correspond to the hard edge analysis in the frozen Laguerre cases by Andraus and Lerner-Brecher and to known results for finite <span><math><mi>β</mi></math></span>.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"313 ","pages":"Article 106229"},"PeriodicalIF":0.6,"publicationDate":"2025-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145578824","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 : 2025-08-19DOI: 10.1016/j.jat.2025.106224
Per G. Nilsson
The aim of this note is to introduce two new generic Banach lattice couples based on weighted spaces of continuous functions, and weighted Radon measures on the positive real-line, denoted by and respectively. This leads to a new approach, based on these couples, of the Sedaev–Semenov result regarding the Calderón–Mityagin property for weighted spaces. As a consequence is obtained a formal equivalence between the concept of divisibility and the relative Calderón–Mityagin Property between and general Banach couples.
{"title":"The Calderón–Mityagin property for couples of weighted Radon measures","authors":"Per G. Nilsson","doi":"10.1016/j.jat.2025.106224","DOIUrl":"10.1016/j.jat.2025.106224","url":null,"abstract":"<div><div>The aim of this note is to introduce two new generic Banach lattice couples based on weighted spaces of continuous functions, and weighted Radon measures on the positive real-line, denoted by <span><math><mover><mrow><mi>C</mi></mrow><mo>⃗</mo></mover></math></span> and <span><math><mover><mrow><mi>M</mi></mrow><mo>⃗</mo></mover></math></span> respectively. This leads to a new approach, based on these couples, of the Sedaev–Semenov result regarding the Calderón–Mityagin property for weighted <span><math><msub><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> spaces. As a consequence is obtained a formal equivalence between the concept of <span><math><mi>K</mi></math></span> divisibility and the relative Calderón–Mityagin Property between <span><math><mover><mrow><mi>M</mi></mrow><mo>⃗</mo></mover></math></span> and general Banach couples.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"314 ","pages":"Article 106224"},"PeriodicalIF":0.6,"publicationDate":"2025-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145683679","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 : 2025-08-19DOI: 10.1016/j.jat.2025.106223
Oleksiy Karlovych, Sandra Mary Thampi
<div><div>Let <span><math><mrow><mi>X</mi><mrow><mo>(</mo><mi>Z</mi><mo>)</mo></mrow></mrow></math></span> be a reflexive rearrangement-invariant Banach sequence space with nontrivial Boyd indices <span><math><mrow><msub><mrow><mi>α</mi></mrow><mrow><mi>X</mi></mrow></msub><mo>,</mo><msub><mrow><mi>β</mi></mrow><mrow><mi>X</mi></mrow></msub></mrow></math></span> and let <span><math><mi>w</mi></math></span> be a symmetric weight in the intersection of the Muckenhoupt classes <span><math><mrow><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn><mo>/</mo><msub><mrow><mi>α</mi></mrow><mrow><mi>X</mi></mrow></msub></mrow></msub><mrow><mo>(</mo><mi>Z</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn><mo>/</mo><msub><mrow><mi>β</mi></mrow><mrow><mi>X</mi></mrow></msub></mrow></msub><mrow><mo>(</mo><mi>Z</mi><mo>)</mo></mrow></mrow></math></span>. Let <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>X</mi><mrow><mo>(</mo><mi>Z</mi><mo>,</mo><mi>w</mi><mo>)</mo></mrow></mrow></msub></math></span> denote the collection of all periodic distributions <span><math><mi>a</mi></math></span> generating bounded Laurent operators <span><math><mrow><mi>L</mi><mrow><mo>(</mo><mi>a</mi><mo>)</mo></mrow></mrow></math></span> on the space <span><math><mrow><mi>X</mi><mrow><mo>(</mo><mi>Z</mi><mo>,</mo><mi>w</mi><mo>)</mo></mrow><mo>=</mo><mrow><mo>{</mo><mi>φ</mi><mo>:</mo><mi>Z</mi><mo>→</mo><mi>ℂ</mi><mo>:</mo><mi>φ</mi><mi>w</mi><mo>∈</mo><mi>X</mi><mrow><mo>(</mo><mi>Z</mi><mo>)</mo></mrow><mo>}</mo></mrow></mrow></math></span>. We show that <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>X</mi><mrow><mo>(</mo><mi>Z</mi><mo>,</mo><mi>w</mi><mo>)</mo></mrow></mrow></msub></math></span> is a Banach algebra. Further, we consider the closure of trigonometric polynomials in <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>X</mi><mrow><mo>(</mo><mi>Z</mi><mo>,</mo><mi>w</mi><mo>)</mo></mrow></mrow></msub></math></span> denoted by <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>X</mi><mrow><mo>(</mo><mi>Z</mi><mo>,</mo><mi>w</mi><mo>)</mo></mrow></mrow></msub></math></span> and <span><math><mrow><msubsup><mrow><mi>H</mi></mrow><mrow><mi>X</mi><mrow><mo>(</mo><mi>Z</mi><mo>,</mo><mi>w</mi><mo>)</mo></mrow></mrow><mrow><mi>∞</mi><mo>,</mo><mo>±</mo></mrow></msubsup><mo>=</mo><mrow><mo>{</mo><mi>a</mi><mo>∈</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>X</mi><mrow><mo>(</mo><mi>Z</mi><mo>,</mo><mi>w</mi><mo>)</mo></mrow></mrow></msub><mo>:</mo><mover><mrow><mi>a</mi></mrow><mrow><mo>̂</mo></mrow></mover><mrow><mo>(</mo><mo>±</mo><mi>n</mi><mo>)</mo></mrow><mo>=</mo><mn>0</mn><mtext> for </mtext><mi>n</mi><mo><</mo><mn>0</mn><mo>}</mo></mrow></mrow></math></span>. We prove that <span><math><mrow><msub><mrow><mi>C</mi></mrow><mrow><mi>X</mi><mrow><mo>(</mo><mi>Z</mi><mo>,</mo><mi>w</mi><mo>)</mo></mrow></mrow></msub><mo>+</mo><msubsup><mrow><mi>H</mi></mrow><mrow><mi>X</mi><mrow><mo>(</mo><mi>Z</mi><mo>,</mo><mi>w<
{"title":"On multiplier analogues of the algebra C+H∞ on weighted rearrangement-invariant sequence spaces","authors":"Oleksiy Karlovych, Sandra Mary Thampi","doi":"10.1016/j.jat.2025.106223","DOIUrl":"10.1016/j.jat.2025.106223","url":null,"abstract":"<div><div>Let <span><math><mrow><mi>X</mi><mrow><mo>(</mo><mi>Z</mi><mo>)</mo></mrow></mrow></math></span> be a reflexive rearrangement-invariant Banach sequence space with nontrivial Boyd indices <span><math><mrow><msub><mrow><mi>α</mi></mrow><mrow><mi>X</mi></mrow></msub><mo>,</mo><msub><mrow><mi>β</mi></mrow><mrow><mi>X</mi></mrow></msub></mrow></math></span> and let <span><math><mi>w</mi></math></span> be a symmetric weight in the intersection of the Muckenhoupt classes <span><math><mrow><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn><mo>/</mo><msub><mrow><mi>α</mi></mrow><mrow><mi>X</mi></mrow></msub></mrow></msub><mrow><mo>(</mo><mi>Z</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn><mo>/</mo><msub><mrow><mi>β</mi></mrow><mrow><mi>X</mi></mrow></msub></mrow></msub><mrow><mo>(</mo><mi>Z</mi><mo>)</mo></mrow></mrow></math></span>. Let <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>X</mi><mrow><mo>(</mo><mi>Z</mi><mo>,</mo><mi>w</mi><mo>)</mo></mrow></mrow></msub></math></span> denote the collection of all periodic distributions <span><math><mi>a</mi></math></span> generating bounded Laurent operators <span><math><mrow><mi>L</mi><mrow><mo>(</mo><mi>a</mi><mo>)</mo></mrow></mrow></math></span> on the space <span><math><mrow><mi>X</mi><mrow><mo>(</mo><mi>Z</mi><mo>,</mo><mi>w</mi><mo>)</mo></mrow><mo>=</mo><mrow><mo>{</mo><mi>φ</mi><mo>:</mo><mi>Z</mi><mo>→</mo><mi>ℂ</mi><mo>:</mo><mi>φ</mi><mi>w</mi><mo>∈</mo><mi>X</mi><mrow><mo>(</mo><mi>Z</mi><mo>)</mo></mrow><mo>}</mo></mrow></mrow></math></span>. We show that <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>X</mi><mrow><mo>(</mo><mi>Z</mi><mo>,</mo><mi>w</mi><mo>)</mo></mrow></mrow></msub></math></span> is a Banach algebra. Further, we consider the closure of trigonometric polynomials in <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>X</mi><mrow><mo>(</mo><mi>Z</mi><mo>,</mo><mi>w</mi><mo>)</mo></mrow></mrow></msub></math></span> denoted by <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>X</mi><mrow><mo>(</mo><mi>Z</mi><mo>,</mo><mi>w</mi><mo>)</mo></mrow></mrow></msub></math></span> and <span><math><mrow><msubsup><mrow><mi>H</mi></mrow><mrow><mi>X</mi><mrow><mo>(</mo><mi>Z</mi><mo>,</mo><mi>w</mi><mo>)</mo></mrow></mrow><mrow><mi>∞</mi><mo>,</mo><mo>±</mo></mrow></msubsup><mo>=</mo><mrow><mo>{</mo><mi>a</mi><mo>∈</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>X</mi><mrow><mo>(</mo><mi>Z</mi><mo>,</mo><mi>w</mi><mo>)</mo></mrow></mrow></msub><mo>:</mo><mover><mrow><mi>a</mi></mrow><mrow><mo>̂</mo></mrow></mover><mrow><mo>(</mo><mo>±</mo><mi>n</mi><mo>)</mo></mrow><mo>=</mo><mn>0</mn><mtext> for </mtext><mi>n</mi><mo><</mo><mn>0</mn><mo>}</mo></mrow></mrow></math></span>. We prove that <span><math><mrow><msub><mrow><mi>C</mi></mrow><mrow><mi>X</mi><mrow><mo>(</mo><mi>Z</mi><mo>,</mo><mi>w</mi><mo>)</mo></mrow></mrow></msub><mo>+</mo><msubsup><mrow><mi>H</mi></mrow><mrow><mi>X</mi><mrow><mo>(</mo><mi>Z</mi><mo>,</mo><mi>w<","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"314 ","pages":"Article 106223"},"PeriodicalIF":0.6,"publicationDate":"2025-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145683609","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 : 2025-08-19DOI: 10.1016/j.jat.2025.106226
Valery Volchkov, Vitaly Volchkov
We investigate some interpolation problems for functions with vanishing integrals over all balls in () of fixed radius. In the case when the set of interpolation nodes is finite a complete solution of the multiple interpolation problem is obtained. In addition, the case when the set of interpolation nodes is infinite and not contained on some line in is studied for the first time. We give a sufficient condition for the solvability of interpolation problem when the set of interpolation nodes is quite rare. We note that the results of the paper are false in the case of dimension .
{"title":"Interpolation of functions with zero integrals over balls of fixed radius","authors":"Valery Volchkov, Vitaly Volchkov","doi":"10.1016/j.jat.2025.106226","DOIUrl":"10.1016/j.jat.2025.106226","url":null,"abstract":"<div><div>We investigate some interpolation problems for functions with vanishing integrals over all balls in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>\u0000 (<span><math><mrow><mi>n</mi><mo>≥</mo><mn>2</mn></mrow></math></span>) of fixed radius. In the case when the set of interpolation nodes is finite a complete solution of the multiple interpolation problem is obtained. In addition, the case when the set of interpolation nodes is infinite and not contained on some line in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> is studied for the first time. We give a sufficient condition for the solvability of interpolation problem when the set of interpolation nodes is quite rare. We note that the results of the paper are false in the case of dimension <span><math><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow></math></span>.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"314 ","pages":"Article 106226"},"PeriodicalIF":0.6,"publicationDate":"2025-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145683673","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 : 2025-08-19DOI: 10.1016/j.jat.2025.106219
Tamás Erdélyi
We show that if is a sequence of numbers with , and then is NOT an ultraflat sequence of unimodular polynomials. This answers a question raised by Zachary Chase.
{"title":"The sequence of partial sums of a unimodular power series is not ultraflat","authors":"Tamás Erdélyi","doi":"10.1016/j.jat.2025.106219","DOIUrl":"10.1016/j.jat.2025.106219","url":null,"abstract":"<div><div>We show that if <span><math><msubsup><mrow><mrow><mo>(</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>)</mo></mrow></mrow><mrow><mi>j</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>∞</mi></mrow></msubsup></math></span> is a sequence of numbers <span><math><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>∈</mo><mi>ℂ</mi></mrow></math></span> with <span><math><mrow><mrow><mo>|</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>|</mo></mrow><mo>=</mo><mn>1</mn></mrow></math></span>, and <span><span><span><math><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>=</mo><munderover><mrow><mo>∑</mo></mrow><mrow><mi>j</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>n</mi></mrow></munderover><msub><mrow><mi>a</mi></mrow><mrow><mi>j</mi></mrow></msub><msup><mrow><mi>z</mi></mrow><mrow><mi>j</mi></mrow></msup><mspace></mspace><mo>,</mo><mspace></mspace><mi>n</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mspace></mspace><mo>,</mo></mrow></math></span></span></span>then <span><math><mrow><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow></math></span> is NOT an ultraflat sequence of unimodular polynomials. This answers a question raised by Zachary Chase.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"313 ","pages":"Article 106219"},"PeriodicalIF":0.6,"publicationDate":"2025-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144908604","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 : 2025-08-19DOI: 10.1016/j.jat.2025.106228
T.M. Dunster
Asymptotic expansions are derived for associated Legendre functions of degree and order , where one or the other of the parameters is large. The expansions are uniformly valid for unbounded real and complex values of the argument , including the singularity . The cases where and are real or purely imaginary are included, which includes conical functions. The approximations involve either exponential or modified Bessel functions, along with slowly varying coefficient functions. The coefficients of the new asymptotic expansions are simple and readily obtained explicitly, allowing for computation to a high degree of accuracy. The results are constructed and rigorously established by employing certain Liouville–Green type expansions where the coefficients appear in the exponent of an exponential function.
{"title":"Simplified uniform asymptotic expansions for associated Legendre and conical functions","authors":"T.M. Dunster","doi":"10.1016/j.jat.2025.106228","DOIUrl":"10.1016/j.jat.2025.106228","url":null,"abstract":"<div><div>Asymptotic expansions are derived for associated Legendre functions of degree <span><math><mi>ν</mi></math></span> and order <span><math><mi>μ</mi></math></span>, where one or the other of the parameters is large. The expansions are uniformly valid for unbounded real and complex values of the argument <span><math><mi>z</mi></math></span>, including the singularity <span><math><mrow><mi>z</mi><mo>=</mo><mn>1</mn></mrow></math></span>. The cases where <span><math><mrow><mi>ν</mi><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></math></span> and <span><math><mi>μ</mi></math></span> are real or purely imaginary are included, which includes conical functions. The approximations involve either exponential or modified Bessel functions, along with slowly varying coefficient functions. The coefficients of the new asymptotic expansions are simple and readily obtained explicitly, allowing for computation to a high degree of accuracy. The results are constructed and rigorously established by employing certain Liouville–Green type expansions where the coefficients appear in the exponent of an exponential function.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"313 ","pages":"Article 106228"},"PeriodicalIF":0.6,"publicationDate":"2025-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145578827","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 : 2025-08-19DOI: 10.1016/j.jat.2025.106225
R. Trigub , V. Volchkov
What is best approximation of a non-integer number by polynomials of degree at most with integer coefficients on the segment in the uniform metric? In the paper, this old problem is solved for any rational number and a segment on the real line. The same problem is solved when the segment is replaced by a disk in and a cube in , both non containing integer points. Best approximation of rational numbers by polynomials with natural coefficients is considered as well. At the same time, the question of uniqueness and non-uniqueness of best approximation polynomials has also been studied. In addition, connection between theorems on best approximation to functions by polynomials with integer coefficients and integer transfinite diameter is established.
{"title":"Best approximation of constants by polynomials with integer coefficients","authors":"R. Trigub , V. Volchkov","doi":"10.1016/j.jat.2025.106225","DOIUrl":"10.1016/j.jat.2025.106225","url":null,"abstract":"<div><div>What is best approximation of a non-integer number <span><math><mrow><mi>λ</mi><mo>∈</mo><mi>R</mi></mrow></math></span> by polynomials <span><math><msub><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> of degree at most <span><math><mi>n</mi></math></span> with integer coefficients on the segment <span><math><mrow><mrow><mo>[</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>]</mo></mrow><mo>⊂</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span> in the uniform metric? In the paper, this old problem is solved for any rational number <span><math><mrow><mi>λ</mi><mo>=</mo><mfrac><mrow><mi>p</mi></mrow><mrow><mi>q</mi></mrow></mfrac></mrow></math></span> and a segment on the real line. The same problem is solved when the segment is replaced by a disk in <span><math><mi>ℂ</mi></math></span> and a cube in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>m</mi></mrow></msup></math></span>, both non containing integer points. Best approximation of rational numbers by polynomials with natural coefficients is considered as well. At the same time, the question of uniqueness and non-uniqueness of best approximation polynomials has also been studied. In addition, connection between theorems on best approximation to functions by polynomials with integer coefficients and integer transfinite diameter is established.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"314 ","pages":"Article 106225"},"PeriodicalIF":0.6,"publicationDate":"2025-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145683672","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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