{"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}
{"title":"Nyström subsampling for functional linear regression","authors":"Jun Fan , Jiading Liu , Lei Shi","doi":"10.1016/j.jat.2025.106176","DOIUrl":"10.1016/j.jat.2025.106176","url":null,"abstract":"<div><div>Kernel methods have proven to be highly effective for functional data analysis, demonstrating significant theoretical and practical success over the past two decades. However, their computational complexity and storage requirements hinder their direct application to large-scale functional data learning problems. In this paper, we address this limitation by investigating the theoretical properties of the Nyström subsampling method within the framework of the functional linear regression model and reproducing kernel Hilbert space. Our proposed algorithm not only overcomes the computational challenges but also achieves the minimax optimal rate of convergence for the excess prediction risk, provided an appropriate subsampling size is chosen. Our error analysis relies on the approximation of integral operators induced by the reproducing kernel and covariance function.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"310 ","pages":"Article 106176"},"PeriodicalIF":0.9,"publicationDate":"2025-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143854851","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":"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":"2025-03-26","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}
{"title":"Rearrangement-invariant norm inequalities for convolution operators","authors":"Ron Kerman , S. Spektor","doi":"10.1016/j.jat.2025.106173","DOIUrl":"10.1016/j.jat.2025.106173","url":null,"abstract":"<div><div>Let <span><math><mrow><mi>k</mi><mo>∈</mo><mrow><mo>(</mo><msub><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>∞</mi></mrow></msub><mo>)</mo></mrow><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span>, where, as usual, <span><math><mrow><msub><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msub><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span> denotes the class of Lebesgue-integrable functions on <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> and <span><math><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>∞</mi></mrow></msub><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span> denotes the class of functions on <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> that are Lebesgue-measurable and bounded almost everywhere. Given <span><math><mrow><mi>f</mi><mo>∈</mo><mrow><mo>(</mo><msub><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∩</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>∞</mi></mrow></msub><mo>)</mo></mrow><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span>, set <span><span><span><math><mrow><mrow><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>k</mi></mrow></msub><mi>f</mi><mo>)</mo></mrow><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msub><mrow><mo>∫</mo></mrow><mrow><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub><mi>k</mi><mrow><mo>(</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo>)</mo></mrow><mi>f</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mspace></mspace><mi>d</mi><mi>y</mi><mo>,</mo><mi>x</mi><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>.</mo></mrow></math></span></span></span>We study inequalities of the form <span><span><span><math><mrow><mi>ρ</mi><mrow><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>k</mi></mrow></msub><mi>f</mi><mo>)</mo></mrow><mo>≤</mo><mi>C</mi><mi>σ</mi><mrow><mo>(</mo><mi>f</mi><mo>)</mo></mrow><mo>,</mo></mrow></math></span></span></span>in which <span><math><mrow><mi>C</mi><mo>></mo><mn>0</mn></mrow></math></span> is independent of <span><math><mrow><mi>f</mi><mo>∈</mo><mrow><mo>(</mo><msub><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∩</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>∞</mi></mrow></msub><mo>)</mo></mrow><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span>. The functionals <span><math><mi>ρ</mi></math></span> and <span><math><mi>σ</mi></math></span> are so-called rearrangement-invariant (r.i.) norms on <span><math><mrow><msub><mrow><mi>M</mi></mrow><mrow><mo>+</mo></mrow></msub><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></mrow></mrow></m","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"310 ","pages":"Article 106173"},"PeriodicalIF":0.9,"publicationDate":"2025-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143714999","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":"Asymptotics of Bergman polynomials for domains with reflection-invariant corners","authors":"Erwin Miña-Díaz , Aron Wennman","doi":"10.1016/j.jat.2025.106172","DOIUrl":"10.1016/j.jat.2025.106172","url":null,"abstract":"<div><div>We study the asymptotic behavior of the Bergman orthogonal polynomials <span><math><msubsup><mrow><mrow><mo>(</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow></mrow><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>∞</mi></mrow></msubsup></math></span> for a class of bounded simply connected domains <span><math><mi>D</mi></math></span>. The class is defined by the requirement that conformal maps <span><math><mi>φ</mi></math></span> of <span><math><mi>D</mi></math></span> onto the unit disk extend analytically across the boundary <span><math><mi>L</mi></math></span> of <span><math><mi>D</mi></math></span>, and that <span><math><msup><mrow><mi>φ</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> has a finite number of zeros <span><math><mrow><msub><mrow><mi>z</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>z</mi></mrow><mrow><mi>q</mi></mrow></msub></mrow></math></span> on <span><math><mi>L</mi></math></span>. The boundary <span><math><mi>L</mi></math></span> is then piecewise analytic with corners at the zeros of <span><math><msup><mrow><mi>φ</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span>. A result of Stylianopoulos implies that a Carleman-type strong asymptotic formula for <span><math><msub><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> holds on the exterior domain <span><math><mrow><mi>ℂ</mi><mo>∖</mo><mover><mrow><mi>D</mi></mrow><mo>¯</mo></mover></mrow></math></span>. We prove that the same formula remains valid across <span><math><mrow><mi>L</mi><mo>∖</mo><mrow><mo>{</mo><msub><mrow><mi>z</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>z</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>}</mo></mrow></mrow></math></span> and on a maximal open subset of <span><math><mi>D</mi></math></span>. As a consequence, the only boundary points that attract zeros of <span><math><msub><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> are the corners. This is in stark contrast to the case when <span><math><mi>φ</mi></math></span> fails to admit an analytic extension past <span><math><mi>L</mi></math></span>, since when this happens the zero counting measure of <span><math><msub><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is known to approach the equilibrium measure for <span><math><mi>L</mi></math></span> along suitable subsequences.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"309 ","pages":"Article 106172"},"PeriodicalIF":0.9,"publicationDate":"2025-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143697047","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":"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":"2025-03-21","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}
Markus Hansen , Benjamin Scharf, Cornelia Schneider
{"title":"Relations between Kondratiev spaces and refined localization Triebel–Lizorkin spaces","authors":"Markus Hansen , Benjamin Scharf, Cornelia Schneider","doi":"10.1016/j.jat.2025.106162","DOIUrl":"10.1016/j.jat.2025.106162","url":null,"abstract":"<div><div>We investigate the close relation between certain weighted Sobolev spaces (Kondratiev spaces) and refined localization spaces from Triebel (2006), Triebel (2008). In particular, using a characterization for refined localization spaces from Scharf (2014), we considerably improve an embedding from Hansen (2013). This embedding is of special interest in connection with convergence rates for adaptive approximation schemes.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"310 ","pages":"Article 106162"},"PeriodicalIF":0.9,"publicationDate":"2025-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143680339","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"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":"2025-03-07","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}
{"title":"A note on diffusion limits for stochastic gradient descent","authors":"Alberto Lanconelli, Christopher S.A. Lauria","doi":"10.1016/j.jat.2025.106160","DOIUrl":"10.1016/j.jat.2025.106160","url":null,"abstract":"<div><div>In the machine learning literature stochastic gradient descent has recently been widely discussed for its purported implicit regularization properties. Much of the theory, that attempts to clarify the role of noise in stochastic gradient algorithms, has approximated stochastic gradient descent by a stochastic differential equation with Gaussian noise. We provide a rigorous theoretical justification for this practice that showcases how the Gaussianity of the noise arises naturally.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"309 ","pages":"Article 106160"},"PeriodicalIF":0.9,"publicationDate":"2025-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143511878","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":"Pseudo s-numbers of embeddings of Gaussian weighted Sobolev spaces","authors":"Van Kien Nguyen","doi":"10.1016/j.jat.2025.106159","DOIUrl":"10.1016/j.jat.2025.106159","url":null,"abstract":"<div><div>In this paper, we study the approximation problem for functions in the Gaussian-weighted Sobolev space <span><math><mrow><msubsup><mrow><mi>W</mi></mrow><mrow><mi>p</mi></mrow><mrow><mi>α</mi></mrow></msubsup><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>,</mo><mi>γ</mi><mo>)</mo></mrow></mrow></math></span> of mixed smoothness <span><math><mrow><mi>α</mi><mo>∈</mo><mi>N</mi></mrow></math></span> with error measured in the Gaussian-weighted space <span><math><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>q</mi></mrow></msub><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>,</mo><mi>γ</mi><mo>)</mo></mrow></mrow></math></span>. We obtain the exact asymptotic order of some pseudo <span><math><mi>s</mi></math></span>-numbers for the cases <span><math><mrow><mn>1</mn><mo>≤</mo><mi>q</mi><mo><</mo><mi>p</mi><mo><</mo><mi>∞</mi></mrow></math></span> and <span><math><mrow><mi>p</mi><mo>=</mo><mi>q</mi><mo>=</mo><mn>2</mn></mrow></math></span>. Additionally, we also obtain an upper bound and a lower bound for some pseudo <span><math><mi>s</mi></math></span>-numbers of the embedding of <span><math><mrow><msubsup><mrow><mi>W</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>α</mi></mrow></msubsup><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>,</mo><mi>γ</mi><mo>)</mo></mrow></mrow></math></span> into <span><math><mrow><msubsup><mrow><mi>L</mi></mrow><mrow><mi>∞</mi></mrow><mrow><msqrt><mrow><mi>g</mi></mrow></msqrt></mrow></msubsup><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span>. Our result is an extension of that obtained in Dinh Dũng and Van Kien Nguyen (IMA Journal of Numerical Analysis, 2023) for approximation and Kolmogorov numbers.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"309 ","pages":"Article 106159"},"PeriodicalIF":0.9,"publicationDate":"2025-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143548063","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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