Journal of Approximation Theory最新文献

{"title":"Matrix valued orthogonal polynomials arising from hexagon tilings with 3 × 3-periodic weightings","authors":"Arno B.J. Kuijlaars","doi":"10.1016/j.jat.2025.106202","DOIUrl":"10.1016/j.jat.2025.106202","url":null,"abstract":"<div><div>Matrix valued orthogonal polynomials (MVOP) appear in the study of doubly periodic tiling models. Of particular interest is their limiting behavior as the degree tends to infinity. In recent years, MVOP associated with doubly periodic domino tilings of the Aztec diamond have been successfully analyzed. The MVOP related to doubly periodic lozenge tilings of a hexagon are more complicated. In this paper we focus on a special subclass of hexagon tilings with 3 × 3 periodicity. The special subclass leads to a genus one spectral curve with additional symmetries that allow us to find an equilibrium measure in an external field explicitly. The equilibrium measure gives the asymptotic distribution for the zeros of the determinant of the MVOP. The associated <span><math><mi>g</mi></math></span>-functions appear in the strong asymptotic formula for the MVOP that we obtain from a steepest descent analysis of the Riemann–Hilbert problem for MVOP.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"311 ","pages":"Article 106202"},"PeriodicalIF":0.9,"publicationDate":"2025-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144167857","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":"Bernstein-type inequalities for mean n-valent functions","authors":"Anton Baranov ,&nbsp;Ilgiz Kayumov ,&nbsp;Rachid Zarouf","doi":"10.1016/j.jat.2025.106200","DOIUrl":"10.1016/j.jat.2025.106200","url":null,"abstract":"<div><div>We derive new integral estimates of the derivatives of mean <span><math><mi>n</mi></math></span>-valent functions in the unit disc. Our results develop and complement estimates obtained by E.<!--> <!-->P. Dolzhenko and A.<!--> <!-->A. Pekarskii, as well as recent inequalities obtained by the authors. As an application, we improve some inverse theorems of rational approximation due to Dolzhenko.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"311 ","pages":"Article 106200"},"PeriodicalIF":0.9,"publicationDate":"2025-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144116319","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":"Universal discretization and sparse recovery","authors":"F. Dai ,&nbsp;V. Temlyakov","doi":"10.1016/j.jat.2025.106199","DOIUrl":"10.1016/j.jat.2025.106199","url":null,"abstract":"<div><div>Recently, it was discovered that for a given function class <span><math><mi>F</mi></math></span> the error of best linear recovery in the square norm can be bounded above by the Kolmogorov width of <span><math><mi>F</mi></math></span> in the uniform norm. That analysis is based on deep results in discretization of the square norm of functions from finite-dimensional subspaces. In this paper we show how very recent results on universal discretization of the square norm of functions from a collection of finite-dimensional subspaces lead to a Lebesgue-type inequality between the error of sparse recovery in the square norm provided by the algorithm based on least squares operator and best sparse approximations in the uniform norm with respect to appropriate dictionaries.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"311 ","pages":"Article 106199"},"PeriodicalIF":0.9,"publicationDate":"2025-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144099552","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":"Minimax and maximin problems for sums of translates on the real axis","authors":"T.M. Nikiforova","doi":"10.1016/j.jat.2025.106190","DOIUrl":"10.1016/j.jat.2025.106190","url":null,"abstract":"<div><div>Sums of translates generalize logarithms of weighted algebraic polynomials. The paper presents the solution to the minimax and maximin problems on the real axis for sums of translates. We prove that there is a unique function that is extremal in both problems. The key in our proof is a reduction to the problem on a segment. For this, we work out an analogue of the Mhaskar–Rakhmanov–Saff theorem, too.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"311 ","pages":"Article 106190"},"PeriodicalIF":0.9,"publicationDate":"2025-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143936458","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":"A lower bound for the Lebesgue constant of the Morrow–Patterson points","authors":"Tomasz Beberok","doi":"10.1016/j.jat.2025.106191","DOIUrl":"10.1016/j.jat.2025.106191","url":null,"abstract":"<div><div>The study of interpolation nodes and their associated Lebesgue constants is a cornerstone of numerical analysis, directly influencing the stability and accuracy of polynomial approximations. In this paper, we examine the Morrow–Patterson points, a specific set of interpolation nodes introduced to construct cubature formulas with the minimal number of points in a square for a fixed degree <span><math><mi>n</mi></math></span>. We prove that their Lebesgue constant has minimal rate of growth of at least <span><math><mrow><mi>O</mi><mrow><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span>.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"311 ","pages":"Article 106191"},"PeriodicalIF":0.9,"publicationDate":"2025-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143935524","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 molecular characterizations of variable Triebel–Lizorkin spaces associated with the Hermite operator and its applications","authors":"Qi Sun ,&nbsp;Ciqiang Zhuo","doi":"10.1016/j.jat.2025.106188","DOIUrl":"10.1016/j.jat.2025.106188","url":null,"abstract":"&lt;div&gt;&lt;div&gt;In this article, we introduce inhomogeneous variable Triebel–Lizorkin spaces, &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;⋅&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;⋅&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;⋅&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, associated with the Hermite operator &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;mo&gt;≔&lt;/mo&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;Δ&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msup&gt;&lt;mrow&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;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, where &lt;span&gt;&lt;math&gt;&lt;mi&gt;Δ&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; is the Laplace operator on &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;, and mainly establish the molecular characterization of these spaces. As applications, we obtain some regularity results to fractional Hermite equations &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;Δ&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msup&gt;&lt;mrow&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;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;σ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;Δ&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msup&gt;&lt;mrow&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;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;σ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; where &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;σ&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;mo&gt;,&lt;/mo&gt;&lt;mi&gt;∞&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, and the boundedness of spectral multiplier associated to the operator &lt;span&gt;&lt;math&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; on the variable Triebel–Lizorkin space &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;⋅&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;⋅&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;⋅&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. Furthermore, we explain the relationship between &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;⋅&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;⋅&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;⋅&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; and the variable Triebel–Lizorkin spaces &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;⋅&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;⋅&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;⋅&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"311 ","pages":"Article 106188"},"PeriodicalIF":0.9,"publicationDate":"2025-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143929170","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":"Orthogonal polynomials on the real line generated by the parameter sequences for a given non-single parameter positive chain sequence","authors":"Daniel O. Veronese,&nbsp;Glalco S. Costa","doi":"10.1016/j.jat.2025.106189","DOIUrl":"10.1016/j.jat.2025.106189","url":null,"abstract":"<div><div>In this paper, given a non-single parameter positive chain sequence <span><math><mrow><msubsup><mrow><mrow><mo>{</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>}</mo></mrow></mrow><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>∞</mi></mrow></msubsup><mo>,</mo></mrow></math></span> we use all the non-minimal parameter sequences for <span><math><msubsup><mrow><mrow><mo>{</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>}</mo></mrow></mrow><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>∞</mi></mrow></msubsup></math></span> in order to generate a whole family of sequences of orthogonal polynomials on the real line. For each non-minimal parameter sequence, the orthogonal polynomials and the associated orthogonality measure are obtained. As an application, corresponding quadratic decompositions are explicitly given. Some examples are considered in order to illustrate the results obtained.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"311 ","pages":"Article 106189"},"PeriodicalIF":0.9,"publicationDate":"2025-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143929171","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":"Global rational approximations of functions with factorially divergent asymptotic series","authors":"N. Castillo,&nbsp;O. Costin,&nbsp;R.D. Costin","doi":"10.1016/j.jat.2025.106178","DOIUrl":"10.1016/j.jat.2025.106178","url":null,"abstract":"<div><div>We construct a new type of convergent, and asymptotic, representations, dyadic expansions. Their convergence is geometric and the region of convergence often extends from infinity down to <span><math><msup><mrow><mn>0</mn></mrow><mrow><mo>+</mo></mrow></msup></math></span>. We show that dyadic expansions are numerically efficient representations.</div><div>For special functions such as Bessel, Airy, Ei, erfc, Gamma, etc. the region of convergence of dyadic series is the complex plane minus a ray, with this cut chosen at will. Dyadic expansions thus provide uniform, geometrically convergent asymptotic expansions including near antistokes rays.</div><div>We prove that relatively general functions, Écalle resurgent ones, possess convergent dyadic expansions.</div><div>These expansions extend to operators, resulting in representations of the resolvent of self-adjoint operators as series in terms of the associated unitary evolution operator evaluated at some prescribed discrete times (alternatively, for positive operators, in terms of the generated semigroup).</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"311 ","pages":"Article 106178"},"PeriodicalIF":0.9,"publicationDate":"2025-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143907604","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":"Approximating positive homogeneous functions with scale invariant neural networks","authors":"Stefan Bamberger ,&nbsp;Reinhard Heckel ,&nbsp;Felix Krahmer","doi":"10.1016/j.jat.2025.106177","DOIUrl":"10.1016/j.jat.2025.106177","url":null,"abstract":"<div><div>We investigate the approximation of positive homogeneous functions, i.e., functions <span><math><mi>f</mi></math></span> satisfying <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>λ</mi><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mi>λ</mi><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span> for all <span><math><mrow><mi>λ</mi><mo>≥</mo><mn>0</mn></mrow></math></span>, with neural networks. Extending previous work, we establish new results explaining under which conditions such functions can be approximated with neural networks. As a key application for this, we analyze to what extent it is possible to solve linear inverse problems with <span><math><mo>ReLu</mo></math></span> networks. Due to the scaling invariance arising from the linearity, an optimal reconstruction function for such a problem is positive homogeneous. In a <span><math><mo>ReLu</mo></math></span> network, this condition translates to considering networks without bias terms. For the recovery of sparse vectors from few linear measurements, our results imply that <span><math><mo>ReLu</mo></math></span> networks with two hidden layers allow approximate recovery with arbitrary precision and arbitrary sparsity level <span><math><mi>s</mi></math></span> in a stable way. In contrast, we also show that with only one hidden layer such networks cannot even recover 1-sparse vectors, not even approximately, and regardless of the width of the network. These findings even apply to a wider class of recovery problems including low-rank matrix recovery and phase retrieval. Our results also shed some light on the seeming contradiction between previous works showing that neural networks for inverse problems typically have very large Lipschitz constants, but still perform very well also for adversarial noise. Namely, the error bounds in our expressivity results include a combination of a small constant term and a term that is linear in the noise level, indicating that robustness issues may occur only for very small noise levels.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"311 ","pages":"Article 106177"},"PeriodicalIF":0.9,"publicationDate":"2025-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143898393","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 a lemma by Brézis and Haraux","authors":"Minh N. Bùi","doi":"10.1016/j.jat.2025.106175","DOIUrl":"10.1016/j.jat.2025.106175","url":null,"abstract":"<div><div>We propose several applications of an often overlooked part of the 1976 paper by Brézis and Haraux, in which the Brézis–Haraux theorem was established. Our results unify and extend various existing ones on the range of a linearly composite monotone operator and provide new insight into their seminal paper.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"311 ","pages":"Article 106175"},"PeriodicalIF":0.9,"publicationDate":"2025-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143882922","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}