Pub Date : 2025-09-01Epub Date: 2025-03-18DOI: 10.1016/j.jat.2025.106162
Markus Hansen , Benjamin Scharf, Cornelia Schneider
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.
{"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":"2025-09-01","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}
Pub Date : 2025-09-01Epub Date: 2025-03-26DOI: 10.1016/j.jat.2025.106173
Ron Kerman , S. Spektor
<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
设k∈(L1+L∞)(Rn)，其中，通常，L1（Rn）表示Rn上的勒贝格可积函数类，L∞(Rn)表示Rn上的几乎处处勒贝格可测且有界的函数类。鉴于f∈(L1∩L∞)(Rn)组(Tkf) (x) =∫Rnk (x−y) f dy (y), x∈Rn。我们研究了ρ(Tkf)≤Cσ(f)的不等式，其中C>；0与f∈(L1∩L∞)（Rn）无关。泛函ρ和σ是所谓的M+(Rn)上的重排不变（r.i）范数，这是Rn上的一类非负可测函数。在r.i.范数的一般情况下首先证明的结果在Orlicz范数的特殊情况下都是专门化和扩展的。
{"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":"2025-09-01","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}
Pub Date : 2025-09-01Epub Date: 2025-04-16DOI: 10.1016/j.jat.2025.106176
Jun Fan , Jiading Liu , Lei Shi
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.
{"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":"2025-09-01","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}
Pub Date : 2025-08-01Epub Date: 2025-02-27DOI: 10.1016/j.jat.2025.106160
Alberto Lanconelli, Christopher S.A. Lauria
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.
{"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":"2025-08-01","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}
Pub Date : 2025-08-01Epub Date: 2025-02-13DOI: 10.1016/j.jat.2025.106150
Chelo Ferreira , José L. López , Ester Pérez Sinusía
We consider the Pearcey integral for large values of and bounded values of . The standard saddle point analysis is difficult to apply because the Pearcey integral is highly oscillating in this region. To overcome this problem we use the modified saddle point method introduced in López et al. (2009). A complete asymptotic analysis is possible with this method, and we derive a complete asymptotic expansion of for large , accompanied by the exact location of the Stokes lines. There are two Stokes lines that divide the complex plane in two different sectors in which behaves differently when is large. The asymptotic approximation is the sum of two asymptotic series whose terms are elementary functions of and . Both of them are of Poincaré type; one of them is given in terms of inverse powers of ; the other one in terms of inverse powers of , and it is multiplied by an exponential factor that behaves differently in the two mentioned sectors. Some numerical experiments illustrate the accuracy of the approximation.
我们考虑的是 Pearcey 积分 P(x,y)，适用于 |x| 的大值和 |y| 的有界值。标准的鞍点分析难以应用，因为皮尔斯积分在这一区域高度振荡。为了克服这个问题，我们采用了 López 等人（2009 年）提出的修正鞍点方法。我们得出了 P(x,y) 在大|x|时的完整渐近展开，并给出了斯托克斯线的精确位置。有两条斯托克斯线将复 x 平面划分为两个不同的扇形区域，当 |x| 较大时，P(x,y) 在这两个扇形区域的表现不同。近似值是两个近似级数之和，其项是 x 和 y 的初等函数。这两个近似级数都是 Poincaré 类型；其中一个用 x 的反幂表示，另一个用 x1/2 的反幂表示，并乘以一个指数因子，在上述两个扇形中表现不同。一些数值实验说明了近似值的准确性。
{"title":"The Pearcey integral in the highly oscillatory region II","authors":"Chelo Ferreira , José L. López , Ester Pérez Sinusía","doi":"10.1016/j.jat.2025.106150","DOIUrl":"10.1016/j.jat.2025.106150","url":null,"abstract":"<div><div>We consider the Pearcey integral <span><math><mrow><mi>P</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></mrow></math></span> for large values of <span><math><mrow><mo>|</mo><mi>x</mi><mo>|</mo></mrow></math></span> and bounded values of <span><math><mrow><mo>|</mo><mi>y</mi><mo>|</mo></mrow></math></span>. The standard saddle point analysis is difficult to apply because the Pearcey integral is highly oscillating in this region. To overcome this problem we use the modified saddle point method introduced in López et al. (2009). A complete asymptotic analysis is possible with this method, and we derive a complete asymptotic expansion of <span><math><mrow><mi>P</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></mrow></math></span> for large <span><math><mrow><mo>|</mo><mi>x</mi><mo>|</mo></mrow></math></span>, accompanied by the exact location of the Stokes lines. There are two Stokes lines that divide the complex <span><math><mrow><mi>x</mi><mo>−</mo></mrow></math></span>plane in two different sectors in which <span><math><mrow><mi>P</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></mrow></math></span> behaves differently when <span><math><mrow><mo>|</mo><mi>x</mi><mo>|</mo></mrow></math></span> is large. The asymptotic approximation is the sum of two asymptotic series whose terms are elementary functions of <span><math><mi>x</mi></math></span> and <span><math><mi>y</mi></math></span>. Both of them are of Poincaré type; one of them is given in terms of inverse powers of <span><math><mi>x</mi></math></span>; the other one in terms of inverse powers of <span><math><msup><mrow><mi>x</mi></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></math></span>, and it is multiplied by an exponential factor that behaves differently in the two mentioned sectors. Some numerical experiments illustrate the accuracy of the approximation.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"309 ","pages":"Article 106150"},"PeriodicalIF":0.9,"publicationDate":"2025-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143445207","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-08-01Epub Date: 2025-02-18DOI: 10.1016/j.jat.2025.106158
A. López García , G. López Lagomasino
We study the relative asymptotics of two sequences of multiple orthogonal polynomials corresponding to two Nikishin systems of measures on the real line, the second one of which is obtained from the first one perturbing the generating measures with non-negative integrable functions. Each Nikishin system consists of two measures.
{"title":"Relative asymptotics of multiple orthogonal polynomials for Nikishin systems of two measures","authors":"A. López García , G. López Lagomasino","doi":"10.1016/j.jat.2025.106158","DOIUrl":"10.1016/j.jat.2025.106158","url":null,"abstract":"<div><div>We study the relative asymptotics of two sequences of multiple orthogonal polynomials corresponding to two Nikishin systems of measures on the real line, the second one of which is obtained from the first one perturbing the generating measures with non-negative integrable functions. Each Nikishin system consists of two measures.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"309 ","pages":"Article 106158"},"PeriodicalIF":0.9,"publicationDate":"2025-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143445208","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-08-01Epub Date: 2025-03-21DOI: 10.1016/j.jat.2025.106172
Erwin Miña-Díaz , Aron Wennman
We study the asymptotic behavior of the Bergman orthogonal polynomials for a class of bounded simply connected domains . The class is defined by the requirement that conformal maps of onto the unit disk extend analytically across the boundary of , and that has a finite number of zeros on . The boundary is then piecewise analytic with corners at the zeros of . A result of Stylianopoulos implies that a Carleman-type strong asymptotic formula for holds on the exterior domain . We prove that the same formula remains valid across and on a maximal open subset of . As a consequence, the only boundary points that attract zeros of are the corners. This is in stark contrast to the case when fails to admit an analytic extension past , since when this happens the zero counting measure of is known to approach the equilibrium measure for along suitable subsequences.
{"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":"2025-08-01","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}
Pub Date : 2025-08-01Epub Date: 2025-02-27DOI: 10.1016/j.jat.2025.106159
Van Kien Nguyen
In this paper, we study the approximation problem for functions in the Gaussian-weighted Sobolev space of mixed smoothness with error measured in the Gaussian-weighted space . We obtain the exact asymptotic order of some pseudo -numbers for the cases and . Additionally, we also obtain an upper bound and a lower bound for some pseudo -numbers of the embedding of into . 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.
本文研究了混合光滑性α∈N的高斯加权Sobolev空间Wpα(Rd,γ)中函数的近似问题，其误差在高斯加权空间Lq（Rd,γ）中测量。在1≤q<；p<；∞且p=q=2的情况下，我们得到了一些伪s数的精确渐近阶。此外，我们还得到了W2α(Rd,γ)嵌入L∞g（Rd）的一些伪s数的上界和下界。我们的结果是Dinh Dũng和Van Kien Nguyen （IMA Journal of Numerical Analysis, 2023）关于近似和Kolmogorov数所得结果的推广。
{"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":"2025-08-01","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}
Pub Date : 2025-06-01Epub Date: 2025-02-06DOI: 10.1016/j.jat.2025.106148
Sung-Soo Byun , Kohei Noda
Non-Hermitian Wishart matrices were introduced in the context of quantum chromodynamics with a baryon chemical potential. These provide chiral extensions of the elliptic Ginibre ensembles as well as non-Hermitian extensions of the classical Wishart/Laguerre ensembles. In this work, we investigate eigenvalues of non-Hermitian Wishart matrices in the symmetry classes of complex and symplectic Ginibre ensembles. We introduce a generalised Christoffel–Darboux formula in the form of a certain second-order differential equation, offering a unified and robust method for analysing correlation functions across all scaling regimes in the model. By employing this method, we derive bulk and edge scaling limits for eigenvalue correlations at both strong and weak non-Hermiticity.
{"title":"Scaling limits of complex and symplectic non-Hermitian Wishart ensembles","authors":"Sung-Soo Byun , Kohei Noda","doi":"10.1016/j.jat.2025.106148","DOIUrl":"10.1016/j.jat.2025.106148","url":null,"abstract":"<div><div>Non-Hermitian Wishart matrices were introduced in the context of quantum chromodynamics with a baryon chemical potential. These provide chiral extensions of the elliptic Ginibre ensembles as well as non-Hermitian extensions of the classical Wishart/Laguerre ensembles. In this work, we investigate eigenvalues of non-Hermitian Wishart matrices in the symmetry classes of complex and symplectic Ginibre ensembles. We introduce a generalised Christoffel–Darboux formula in the form of a certain second-order differential equation, offering a unified and robust method for analysing correlation functions across all scaling regimes in the model. By employing this method, we derive bulk and edge scaling limits for eigenvalue correlations at both strong and weak non-Hermiticity.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"308 ","pages":"Article 106148"},"PeriodicalIF":0.9,"publicationDate":"2025-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143348474","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 : 2025-06-01Epub Date: 2025-02-07DOI: 10.1016/j.jat.2025.106145
Carlos Beltrán , Damir Ferizović , Pedro R. López-Gómez
We construct measure-preserving mappings from the -dimensional unit cube to the -dimensional unit ball and the compact rank one symmetric spaces, namely the -dimensional sphere, the real, complex, and quaternionic projective spaces, and the Cayley plane. We also give a procedure to generate measure-preserving mappings from the -dimensional unit cube to product spaces and fiber bundles under certain conditions.
{"title":"Measure-preserving mappings from the unit cube to some symmetric spaces","authors":"Carlos Beltrán , Damir Ferizović , Pedro R. López-Gómez","doi":"10.1016/j.jat.2025.106145","DOIUrl":"10.1016/j.jat.2025.106145","url":null,"abstract":"<div><div>We construct measure-preserving mappings from the <span><math><mi>d</mi></math></span>-dimensional unit cube to the <span><math><mi>d</mi></math></span>-dimensional unit ball and the compact rank one symmetric spaces, namely the <span><math><mi>d</mi></math></span>-dimensional sphere, the real, complex, and quaternionic projective spaces, and the Cayley plane. We also give a procedure to generate measure-preserving mappings from the <span><math><mi>d</mi></math></span>-dimensional unit cube to product spaces and fiber bundles under certain conditions.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"308 ","pages":"Article 106145"},"PeriodicalIF":0.9,"publicationDate":"2025-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143379078","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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