Pub Date : 2025-11-01Epub Date: 2025-04-18DOI: 10.1016/j.jat.2025.106175
Minh N. Bùi
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.
{"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":"2025-11-01","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}
Pub Date : 2025-11-01Epub Date: 2025-05-26DOI: 10.1016/j.jat.2025.106202
Arno B.J. Kuijlaars
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 -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.
{"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":"2025-11-01","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}
Pub Date : 2025-11-01Epub Date: 2025-04-23DOI: 10.1016/j.jat.2025.106178
N. Castillo, O. Costin, R.D. Costin
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 . We show that dyadic expansions are numerically efficient representations.
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.
We prove that relatively general functions, Écalle resurgent ones, possess convergent dyadic expansions.
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).
我们构造了一种新的收敛的，渐近的，表示，二进展开式。它们的收敛是几何的，收敛的区域通常从无穷远处延伸到0+。我们证明了二进展开式是数值上有效的表示。对于特殊的函数，如Bessel， Airy, Ei, erfc， Gamma等，并矢级数的收敛区域是复平面减去一条射线，这个切割可以随意选择。因此，并矢展开式提供了均匀的、几何收敛的渐近展开式，包括近反斯托克斯射线。我们证明了相对一般的函数Écalle复活函数具有收敛的二进展开式。这些展开式扩展到算子，导致自伴随算子的解表示为在某些规定的离散时间内计算的相关的酉演化算子的级数（或者，对于正算子，根据生成的半群）。
{"title":"Global rational approximations of functions with factorially divergent asymptotic series","authors":"N. Castillo, O. Costin, 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":"2025-11-01","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}
Pub Date : 2025-11-01Epub Date: 2025-05-06DOI: 10.1016/j.jat.2025.106188
Qi Sun , Ciqiang Zhuo
<div><div>In this article, we introduce inhomogeneous variable Triebel–Lizorkin spaces, <span><math><mrow><msubsup><mrow><mi>F</mi></mrow><mrow><mi>p</mi><mrow><mo>(</mo><mi>⋅</mi><mo>)</mo></mrow><mo>,</mo><mi>q</mi><mrow><mo>(</mo><mi>⋅</mi><mo>)</mo></mrow></mrow><mrow><mi>α</mi><mrow><mo>(</mo><mi>⋅</mi><mo>)</mo></mrow><mo>,</mo><mi>H</mi></mrow></msubsup><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span>, associated with the Hermite operator <span><math><mrow><mi>H</mi><mo>≔</mo><mo>−</mo><mi>Δ</mi><mo>+</mo><msup><mrow><mrow><mo>|</mo><mi>x</mi><mo>|</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup></mrow></math></span>, where <span><math><mi>Δ</mi></math></span> is the Laplace operator on <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>, and mainly establish the molecular characterization of these spaces. As applications, we obtain some regularity results to fractional Hermite equations <span><math><mrow><msup><mrow><mrow><mo>(</mo><mo>−</mo><mi>Δ</mi><mo>+</mo><msup><mrow><mrow><mo>|</mo><mi>x</mi><mo>|</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow></mrow><mrow><mi>σ</mi></mrow></msup><mi>u</mi><mo>=</mo><mi>f</mi><mo>,</mo><mspace></mspace><msup><mrow><mrow><mo>(</mo><mo>−</mo><mi>Δ</mi><mo>+</mo><msup><mrow><mrow><mo>|</mo><mi>x</mi><mo>|</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mi>I</mi><mo>)</mo></mrow></mrow><mrow><mi>σ</mi></mrow></msup><mi>u</mi><mo>=</mo><mi>f</mi><mo>,</mo></mrow></math></span> where <span><math><mrow><mi>σ</mi><mo>∈</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow></mrow></math></span>, and the boundedness of spectral multiplier associated to the operator <span><math><mi>H</mi></math></span> on the variable Triebel–Lizorkin space <span><math><mrow><msubsup><mrow><mi>F</mi></mrow><mrow><mi>p</mi><mrow><mo>(</mo><mi>⋅</mi><mo>)</mo></mrow><mo>,</mo><mi>q</mi><mrow><mo>(</mo><mi>⋅</mi><mo>)</mo></mrow></mrow><mrow><mi>α</mi><mrow><mo>(</mo><mi>⋅</mi><mo>)</mo></mrow><mo>,</mo><mi>H</mi></mrow></msubsup><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span>. Furthermore, we explain the relationship between <span><math><mrow><msubsup><mrow><mi>F</mi></mrow><mrow><mi>p</mi><mrow><mo>(</mo><mi>⋅</mi><mo>)</mo></mrow><mo>,</mo><mi>q</mi><mrow><mo>(</mo><mi>⋅</mi><mo>)</mo></mrow></mrow><mrow><mi>α</mi><mrow><mo>(</mo><mi>⋅</mi><mo>)</mo></mrow><mo>,</mo><mi>H</mi></mrow></msubsup><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span> and the variable Triebel–Lizorkin spaces <span><math><mrow><msubsup><mrow><mi>F</mi></mrow><mrow><mi>p</mi><mrow><mo>(</mo><mi>⋅</mi><mo>)</mo></mrow><mo>,</mo><mi>q</mi><mrow><mo>(</mo><mi>⋅</mi><mo>)</mo></mrow></mrow><mrow><mi>α</mi><mrow><mo>(</mo><mi>⋅</mi><mo>)</mo></mrow></mrow></msubsup><mrow><mo>(</mo><msup><mrow>
本文引入了非齐次变量triiebel - lizorkin空间Fp(⋅),q(⋅)α(⋅)，H(Rn)，并结合Hermite算子H（Δ+|x|2），其中Δ为Rn上的拉普拉斯算子，建立了这些空间的分子表征。作为应用，我们得到了分数阶Hermite方程（−Δ+|x|2）σu=f,(−Δ+|x|2+I)σu=f的一些正则性结果，其中σ∈(0,∞)，以及变量triiebel - lizorkin空间Fp(⋅),q(⋅)α(⋅)，H（Rn）上与算子H相关的谱乘子的有界性。此外，我们通过原子分解解释了Fp（⋅）、q(⋅)α(⋅)、H（Rn）与变量triiebel - lizorkin空间Fp（⋅）、q(⋅)α(⋅)(Rn) （Diening et al.（2009）引入）之间的关系。
{"title":"The molecular characterizations of variable Triebel–Lizorkin spaces associated with the Hermite operator and its applications","authors":"Qi Sun , Ciqiang Zhuo","doi":"10.1016/j.jat.2025.106188","DOIUrl":"10.1016/j.jat.2025.106188","url":null,"abstract":"<div><div>In this article, we introduce inhomogeneous variable Triebel–Lizorkin spaces, <span><math><mrow><msubsup><mrow><mi>F</mi></mrow><mrow><mi>p</mi><mrow><mo>(</mo><mi>⋅</mi><mo>)</mo></mrow><mo>,</mo><mi>q</mi><mrow><mo>(</mo><mi>⋅</mi><mo>)</mo></mrow></mrow><mrow><mi>α</mi><mrow><mo>(</mo><mi>⋅</mi><mo>)</mo></mrow><mo>,</mo><mi>H</mi></mrow></msubsup><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span>, associated with the Hermite operator <span><math><mrow><mi>H</mi><mo>≔</mo><mo>−</mo><mi>Δ</mi><mo>+</mo><msup><mrow><mrow><mo>|</mo><mi>x</mi><mo>|</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup></mrow></math></span>, where <span><math><mi>Δ</mi></math></span> is the Laplace operator on <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>, and mainly establish the molecular characterization of these spaces. As applications, we obtain some regularity results to fractional Hermite equations <span><math><mrow><msup><mrow><mrow><mo>(</mo><mo>−</mo><mi>Δ</mi><mo>+</mo><msup><mrow><mrow><mo>|</mo><mi>x</mi><mo>|</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow></mrow><mrow><mi>σ</mi></mrow></msup><mi>u</mi><mo>=</mo><mi>f</mi><mo>,</mo><mspace></mspace><msup><mrow><mrow><mo>(</mo><mo>−</mo><mi>Δ</mi><mo>+</mo><msup><mrow><mrow><mo>|</mo><mi>x</mi><mo>|</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mi>I</mi><mo>)</mo></mrow></mrow><mrow><mi>σ</mi></mrow></msup><mi>u</mi><mo>=</mo><mi>f</mi><mo>,</mo></mrow></math></span> where <span><math><mrow><mi>σ</mi><mo>∈</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow></mrow></math></span>, and the boundedness of spectral multiplier associated to the operator <span><math><mi>H</mi></math></span> on the variable Triebel–Lizorkin space <span><math><mrow><msubsup><mrow><mi>F</mi></mrow><mrow><mi>p</mi><mrow><mo>(</mo><mi>⋅</mi><mo>)</mo></mrow><mo>,</mo><mi>q</mi><mrow><mo>(</mo><mi>⋅</mi><mo>)</mo></mrow></mrow><mrow><mi>α</mi><mrow><mo>(</mo><mi>⋅</mi><mo>)</mo></mrow><mo>,</mo><mi>H</mi></mrow></msubsup><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span>. Furthermore, we explain the relationship between <span><math><mrow><msubsup><mrow><mi>F</mi></mrow><mrow><mi>p</mi><mrow><mo>(</mo><mi>⋅</mi><mo>)</mo></mrow><mo>,</mo><mi>q</mi><mrow><mo>(</mo><mi>⋅</mi><mo>)</mo></mrow></mrow><mrow><mi>α</mi><mrow><mo>(</mo><mi>⋅</mi><mo>)</mo></mrow><mo>,</mo><mi>H</mi></mrow></msubsup><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span> and the variable Triebel–Lizorkin spaces <span><math><mrow><msubsup><mrow><mi>F</mi></mrow><mrow><mi>p</mi><mrow><mo>(</mo><mi>⋅</mi><mo>)</mo></mrow><mo>,</mo><mi>q</mi><mrow><mo>(</mo><mi>⋅</mi><mo>)</mo></mrow></mrow><mrow><mi>α</mi><mrow><mo>(</mo><mi>⋅</mi><mo>)</mo></mrow></mrow></msubsup><mrow><mo>(</mo><msup><mrow>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"311 ","pages":"Article 106188"},"PeriodicalIF":0.9,"publicationDate":"2025-11-01","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}
Pub Date : 2025-11-01Epub Date: 2025-05-06DOI: 10.1016/j.jat.2025.106189
Daniel O. Veronese, Glalco S. Costa
In this paper, given a non-single parameter positive chain sequence we use all the non-minimal parameter sequences for 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.
{"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, 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":"2025-11-01","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}
Pub Date : 2025-11-01Epub Date: 2025-04-23DOI: 10.1016/j.jat.2025.106177
Stefan Bamberger , Reinhard Heckel , Felix Krahmer
We investigate the approximation of positive homogeneous functions, i.e., functions satisfying for all , 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 networks. Due to the scaling invariance arising from the linearity, an optimal reconstruction function for such a problem is positive homogeneous. In a network, this condition translates to considering networks without bias terms. For the recovery of sparse vectors from few linear measurements, our results imply that networks with two hidden layers allow approximate recovery with arbitrary precision and arbitrary sparsity level 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.
{"title":"Approximating positive homogeneous functions with scale invariant neural networks","authors":"Stefan Bamberger , Reinhard Heckel , 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":"2025-11-01","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}
Pub Date : 2025-11-01Epub Date: 2025-05-13DOI: 10.1016/j.jat.2025.106199
F. Dai , V. Temlyakov
Recently, it was discovered that for a given function class the error of best linear recovery in the square norm can be bounded above by the Kolmogorov width of 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.
{"title":"Universal discretization and sparse recovery","authors":"F. Dai , 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":"2025-11-01","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}
Pub Date : 2025-09-01Epub Date: 2025-03-26DOI: 10.1016/j.jat.2025.106174
Sonia Acinas , Sergio Favier , Rosa Lorenzo
In this paper, we consider the best multivalued polynomial approximation operator for functions in an Orlicz Space . We obtain its characterization involving and , which are the left and right derivative functions of . And then, we extend the operator to . We also get pointwise convergence of this extension, where the Calderón–Zygmund class adapted to plays an important role.
{"title":"Extension of the best polynomial operator in generalized Orlicz Spaces","authors":"Sonia Acinas , Sergio Favier , Rosa Lorenzo","doi":"10.1016/j.jat.2025.106174","DOIUrl":"10.1016/j.jat.2025.106174","url":null,"abstract":"<div><div>In this paper, we consider the best multivalued polynomial approximation operator for functions in an Orlicz Space <span><math><mrow><msup><mrow><mi>L</mi></mrow><mrow><mi>φ</mi></mrow></msup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow></mrow></math></span>. We obtain its characterization involving <span><math><msup><mrow><mi>ψ</mi></mrow><mrow><mo>−</mo></mrow></msup></math></span> and <span><math><msup><mrow><mi>ψ</mi></mrow><mrow><mo>+</mo></mrow></msup></math></span>, which are the left and right derivative functions of <span><math><mi>φ</mi></math></span>. And then, we extend the operator to <span><math><mrow><msup><mrow><mi>L</mi></mrow><mrow><msup><mrow><mi>ψ</mi></mrow><mrow><mo>+</mo></mrow></msup></mrow></msup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow></mrow></math></span>. We also get pointwise convergence of this extension, where the Calderón–Zygmund class <span><math><mrow><msubsup><mrow><mi>t</mi></mrow><mrow><mi>m</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span> adapted to <span><math><mrow><msup><mrow><mi>L</mi></mrow><mrow><msup><mrow><mi>ψ</mi></mrow><mrow><mo>+</mo></mrow></msup></mrow></msup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow></mrow></math></span> plays an important role.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"310 ","pages":"Article 106174"},"PeriodicalIF":0.9,"publicationDate":"2025-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143740051","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 : 2025-09-01Epub Date: 2025-03-07DOI: 10.1016/j.jat.2025.106161
Christophe Charlier
<div><div>We introduce the point process</div><div><span><math><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></mfrac><msub><mrow><mo>∏</mo></mrow><mrow><mn>1</mn><mo>≤</mo><mi>j</mi><mo><</mo><mi>k</mi><mo>≤</mo><mi>n</mi></mrow></msub><msup><mrow><mrow><mo>|</mo><msup><mrow><mi>e</mi></mrow><mrow><mi>i</mi><msub><mrow><mi>θ</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></msup><mo>+</mo><msup><mrow><mi>e</mi></mrow><mrow><mi>i</mi><msub><mrow><mi>θ</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></msup><mo>|</mo></mrow></mrow><mrow><mi>β</mi></mrow></msup><msubsup><mrow><mo>∏</mo></mrow><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></msubsup><mi>d</mi><msub><mrow><mi>θ</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>,</mo><mspace></mspace><msub><mrow><mi>θ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>∈</mo><mrow><mo>(</mo><mo>−</mo><mi>π</mi><mo>,</mo><mi>π</mi><mo>]</mo></mrow><mo>,</mo><mspace></mspace><mi>β</mi><mo>></mo><mn>0</mn><mo>,</mo></mrow></math></span></div><div>where <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is the normalization constant. This point process is <em>attractive</em>: it involves <span><math><mi>n</mi></math></span> dependent, uniformly distributed random variables on the unit circle that attract each other. (For comparison, the well-studied C<span><math><mi>β</mi></math></span>E involves <span><math><mi>n</mi></math></span> uniformly distributed random variables on the unit circle that repel each other.) We consider linear statistics of the form <span><math><mrow><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></msubsup><mi>g</mi><mrow><mo>(</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span> as <span><math><mrow><mi>n</mi><mo>→</mo><mi>∞</mi></mrow></math></span>, where <span><math><mrow><mi>g</mi><mo>∈</mo><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>q</mi></mrow></msup></mrow></math></span> and <span><math><mrow><mn>2</mn><mi>π</mi></mrow></math></span>-periodic. We prove that the leading order fluctuations around the mean are of order <span><math><mi>n</mi></math></span> and given by <span><math><mrow><mrow><mo>(</mo><mrow><mi>g</mi><mrow><mo>(</mo><mi>U</mi><mo>)</mo></mrow><mo>−</mo><msubsup><mrow><mo>∫</mo></mrow><mrow><mo>−</mo><mi>π</mi></mrow><mrow><mi>π</mi></mrow></msubsup><mi>g</mi><mrow><mo>(</mo><mi>θ</mi><mo>)</mo></mrow><mfrac><mrow><mi>d</mi><mi>θ</mi></mrow><mrow><mn>2</mn><mi>π</mi></mrow></mfrac></mrow><mo>)</mo></mrow><mi>n</mi></mrow></math></span>, where <span><math><mrow><mi>U</mi><mo>∼</mo><mi>Uniform</mi><mrow><mo>(</mo><mo>−</mo><mi>π</mi><mo>,</mo><mi>π</mi><mo>]</mo></mrow></mrow></math></span>. We also prove that the subleading fluctuations around the mean are
{"title":"A point process on the unit circle with antipodal interactions","authors":"Christophe Charlier","doi":"10.1016/j.jat.2025.106161","DOIUrl":"10.1016/j.jat.2025.106161","url":null,"abstract":"<div><div>We introduce the point process</div><div><span><math><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></mfrac><msub><mrow><mo>∏</mo></mrow><mrow><mn>1</mn><mo>≤</mo><mi>j</mi><mo><</mo><mi>k</mi><mo>≤</mo><mi>n</mi></mrow></msub><msup><mrow><mrow><mo>|</mo><msup><mrow><mi>e</mi></mrow><mrow><mi>i</mi><msub><mrow><mi>θ</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></msup><mo>+</mo><msup><mrow><mi>e</mi></mrow><mrow><mi>i</mi><msub><mrow><mi>θ</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></msup><mo>|</mo></mrow></mrow><mrow><mi>β</mi></mrow></msup><msubsup><mrow><mo>∏</mo></mrow><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></msubsup><mi>d</mi><msub><mrow><mi>θ</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>,</mo><mspace></mspace><msub><mrow><mi>θ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>∈</mo><mrow><mo>(</mo><mo>−</mo><mi>π</mi><mo>,</mo><mi>π</mi><mo>]</mo></mrow><mo>,</mo><mspace></mspace><mi>β</mi><mo>></mo><mn>0</mn><mo>,</mo></mrow></math></span></div><div>where <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is the normalization constant. This point process is <em>attractive</em>: it involves <span><math><mi>n</mi></math></span> dependent, uniformly distributed random variables on the unit circle that attract each other. (For comparison, the well-studied C<span><math><mi>β</mi></math></span>E involves <span><math><mi>n</mi></math></span> uniformly distributed random variables on the unit circle that repel each other.) We consider linear statistics of the form <span><math><mrow><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></msubsup><mi>g</mi><mrow><mo>(</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span> as <span><math><mrow><mi>n</mi><mo>→</mo><mi>∞</mi></mrow></math></span>, where <span><math><mrow><mi>g</mi><mo>∈</mo><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>q</mi></mrow></msup></mrow></math></span> and <span><math><mrow><mn>2</mn><mi>π</mi></mrow></math></span>-periodic. We prove that the leading order fluctuations around the mean are of order <span><math><mi>n</mi></math></span> and given by <span><math><mrow><mrow><mo>(</mo><mrow><mi>g</mi><mrow><mo>(</mo><mi>U</mi><mo>)</mo></mrow><mo>−</mo><msubsup><mrow><mo>∫</mo></mrow><mrow><mo>−</mo><mi>π</mi></mrow><mrow><mi>π</mi></mrow></msubsup><mi>g</mi><mrow><mo>(</mo><mi>θ</mi><mo>)</mo></mrow><mfrac><mrow><mi>d</mi><mi>θ</mi></mrow><mrow><mn>2</mn><mi>π</mi></mrow></mfrac></mrow><mo>)</mo></mrow><mi>n</mi></mrow></math></span>, where <span><math><mrow><mi>U</mi><mo>∼</mo><mi>Uniform</mi><mrow><mo>(</mo><mo>−</mo><mi>π</mi><mo>,</mo><mi>π</mi><mo>]</mo></mrow></mrow></math></span>. We also prove that the subleading fluctuations around the mean are ","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"310 ","pages":"Article 106161"},"PeriodicalIF":0.9,"publicationDate":"2025-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143577985","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-09-01Epub Date: 2025-03-21DOI: 10.1016/j.jat.2025.106171
Marc Kesseböhmer, Aljoscha Niemann
In this note, we present an abstract approach to study asymptotic orders for adaptive approximations with respect to a monotone set function defined on dyadic cubes. We determine the exact upper order in terms of the critical value of the corresponding -partition function, and we are able to provide upper and lower bounds in terms of fractal-geometric quantities. With properly chosen , our new approach has applications in many different areas of mathematics, including the spectral theory of Kreĭn–Feller operators, quantisation dimensions of compactly supported probability measures, and the exact asymptotic order for Kolmogorov, Gel'fand and linear widths for Sobolev embeddings into the Lebesgue space .
{"title":"Exact asymptotic order for generalised adaptive approximations","authors":"Marc Kesseböhmer, Aljoscha Niemann","doi":"10.1016/j.jat.2025.106171","DOIUrl":"10.1016/j.jat.2025.106171","url":null,"abstract":"<div><div>In this note, we present an abstract approach to study asymptotic orders for adaptive approximations with respect to a monotone set function <span><math><mi>J</mi></math></span> defined on dyadic cubes. We determine the exact upper order in terms of the critical value of the corresponding <span><math><mi>J</mi></math></span>-partition function, and we are able to provide upper and lower bounds in terms of fractal-geometric quantities. With properly chosen <span><math><mi>J</mi></math></span>, our new approach has applications in many different areas of mathematics, including the spectral theory of Kreĭn–Feller operators, quantisation dimensions of compactly supported probability measures, and the exact asymptotic order for Kolmogorov, Gel'fand and linear widths for Sobolev embeddings into the Lebesgue space <span><math><msubsup><mrow><mi>L</mi></mrow><mrow><mi>ν</mi></mrow><mrow><mi>p</mi></mrow></msubsup></math></span>.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"310 ","pages":"Article 106171"},"PeriodicalIF":0.9,"publicationDate":"2025-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143680340","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}
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