Journal of Approximation Theory最新文献

{"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}

{"title":"On multiplier analogues of the algebra C+H∞ on weighted rearrangement-invariant sequence spaces","authors":"Oleksiy Karlovych,&nbsp;Sandra Mary Thampi","doi":"10.1016/j.jat.2025.106223","DOIUrl":"10.1016/j.jat.2025.106223","url":null,"abstract":"&lt;div&gt;&lt;div&gt;Let &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; be a reflexive rearrangement-invariant Banach sequence space with nontrivial Boyd indices &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;β&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; and let &lt;span&gt;&lt;math&gt;&lt;mi&gt;w&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; be a symmetric weight in the intersection of the Muckenhoupt classes &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;β&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. Let &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;w&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; denote the collection of all periodic distributions &lt;span&gt;&lt;math&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; generating bounded Laurent operators &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; on the space &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;w&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mi&gt;φ&lt;/mi&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mo&gt;→&lt;/mo&gt;&lt;mi&gt;ℂ&lt;/mi&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;mi&gt;φ&lt;/mi&gt;&lt;mi&gt;w&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. We show that &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;w&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; is a Banach algebra. Further, we consider the closure of trigonometric polynomials in &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;w&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; denoted by &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;w&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;w&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;∞&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;±&lt;/mo&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;w&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;̂&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mo&gt;±&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mtext&gt; for &lt;/mtext&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. We prove that &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;w&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;w&lt;","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}

{"title":"Interpolation of functions with zero integrals over balls of fixed radius","authors":"Valery Volchkov,&nbsp;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}

{"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}

{"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}

{"title":"Best approximation of constants by polynomials with integer coefficients","authors":"R. Trigub ,&nbsp;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}

{"title":"Comparing the degree of constrained and unconstrained trigonometric approximation","authors":"D. Leviatan ,&nbsp;I. Shevchuk ,&nbsp;V. Shevchuk","doi":"10.1016/j.jat.2025.106220","DOIUrl":"10.1016/j.jat.2025.106220","url":null,"abstract":"<div><div>Let <span><math><mrow><mi>r</mi><mo>,</mo><mi>s</mi><mo>∈</mo><mi>N</mi></mrow></math></span>. For a continuous <span><math><mrow><mn>2</mn><mi>π</mi></mrow></math></span>-periodic function, changing its monotonicity <span><math><mrow><mn>2</mn><mi>s</mi></mrow></math></span> times in a period, and whose degree of approximation by trigonometric polynomials of degree <span><math><mrow><mo>&lt;</mo><mi>n</mi></mrow></math></span>, is <span><math><mrow><mo>≤</mo><msup><mrow><mi>n</mi></mrow><mrow><mo>−</mo><mi>r</mi></mrow></msup></mrow></math></span>, <span><math><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></math></span>, we investigate its degree of approximation by such polynomials that, in addition, follow the changes of monotonicity. Obviously, the unconstrained degree is smaller than the constrained one, but for <span><math><mrow><mi>r</mi><mo>&gt;</mo><mn>2</mn><mi>s</mi><mo>−</mo><mn>2</mn></mrow></math></span>, there is a constant <span><math><mrow><mi>c</mi><mrow><mo>(</mo><mi>s</mi><mo>,</mo><mi>r</mi><mo>)</mo></mrow></mrow></math></span> such that the constrained degree is <span><math><mrow><mo>≤</mo><mi>c</mi><mrow><mo>(</mo><mi>s</mi><mo>,</mo><mi>r</mi><mo>)</mo></mrow><msup><mrow><mi>n</mi></mrow><mrow><mo>−</mo><mi>r</mi></mrow></msup></mrow></math></span>, <span><math><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></math></span>. On the other hand we show that, in general, this is invalid for <span><math><mrow><mi>r</mi><mo>≤</mo><mn>2</mn><mi>s</mi><mo>−</mo><mn>2</mn></mrow></math></span>.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"312 ","pages":"Article 106220"},"PeriodicalIF":0.6,"publicationDate":"2025-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144809532","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":"Polynomial approximation in L2 with the double-sided exponential weight via complex analysis","authors":"Pierre Bizeul ,&nbsp;Boaz Klartag","doi":"10.1016/j.jat.2025.106218","DOIUrl":"10.1016/j.jat.2025.106218","url":null,"abstract":"&lt;div&gt;&lt;div&gt;We study the problem of polynomial approximation in &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;\u0000 (&lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;μ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;), where &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;μ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;\u0000 (&lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;) = &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. We show that for any absolutely continuous function &lt;span&gt;&lt;math&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;munderover&gt;&lt;mrow&gt;&lt;mo&gt;∑&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;∞&lt;/mi&gt;&lt;/mrow&gt;&lt;/munderover&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;log&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mo&gt;〈&lt;/mo&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;〉&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mfenced&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;∫&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;log&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;μ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;∫&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;μ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/mfenced&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; for some universal constant &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, where &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; are the orthonormal polynomials associated with &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;μ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;. This inequality is tight in the sense that &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;log&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; on the left-hand side cannot be replaced by &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;log&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; for any sequence &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;→&lt;/mo&gt;&lt;mi&gt;∞&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. When the right-hand side is bounded, this inequality implies a logarithmic rate of approximati","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"312 ","pages":"Article 106218"},"PeriodicalIF":0.6,"publicationDate":"2025-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144842810","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 positive Jacobi matrices with compact inverses","authors":"Pavel Šťovíček ,&nbsp;Grzegorz Świderski","doi":"10.1016/j.jat.2025.106217","DOIUrl":"10.1016/j.jat.2025.106217","url":null,"abstract":"<div><div>We consider positive Jacobi matrices <span><math><mi>J</mi></math></span> with compact inverses and consequently with purely discrete spectra. A number of properties of the corresponding sequence of orthogonal polynomials is studied including the convergence of their zeros, the vague convergence of the zero counting measures and of the Christoffel–Darboux kernels on the diagonal. Particularly, if the inverse of <span><math><mi>J</mi></math></span> belongs to some Schatten class, we identify the asymptotic behavior of the sequence of orthogonal polynomials and express it in terms of its regularized characteristic function. In the even more special case when the inverse of <span><math><mi>J</mi></math></span> belongs to the trace class, we derive various formulas for the orthogonality measure, eigenvectors of <span><math><mi>J</mi></math></span> as well as for the functions of the second kind and related objects. These general results are given a more explicit form in the case when <span><math><mrow><mo>−</mo><mi>J</mi></mrow></math></span> is a generator of a Birth–Death process. Among others, we provide a formula for the trace of the inverse of <span><math><mi>J</mi></math></span>. We illustrate our results by introducing and studying a modification of the <span><math><mi>q</mi></math></span>-Laguerre polynomials corresponding to a determinate moment problem.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"313 ","pages":"Article 106217"},"PeriodicalIF":0.6,"publicationDate":"2025-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144763703","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":"Construction and approximation properties of exact neural network interpolation operators activated by entire functions","authors":"Dansheng Yu","doi":"10.1016/j.jat.2025.106215","DOIUrl":"10.1016/j.jat.2025.106215","url":null,"abstract":"<div><div>The construction and approximation properties of exact neural network interpolation are the important and challenging topics on approximation by neural networks. Most research on exact neural network interpolation has focused on establishing existence, with very few specifically constructed interpolation neural networks proposed. The main purpose of the present paper is to provide a method for directly constructing exact neural network interpolation operators, which has the advantages that all the components in the neural network operators are explicitly known, such as the coefficients, the weights and the thresholds. By employing some important methods in approximation theory, such as the equivalence between the <span><math><mrow><mi>K</mi><mo>−</mo></mrow></math></span>functional and the modulus of continuity of the function, Berens–Lorentz Lemma, and two useful estimates of the derivatives of the operators, we establish both the direct and the converse results of approximation by the new interpolation operators, and thus obtain an equivalence characterization theorem. We also introduce a type of neural network interpolation operators with four layers and a type of max-product neural network operators, rigorously analyzing their approximation properties.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"312 ","pages":"Article 106215"},"PeriodicalIF":0.6,"publicationDate":"2025-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144721942","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}