Pub 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":"2025-03-07","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}
Pub 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":"2025-02-27","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}
Pub 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":"2025-02-27","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}
Pub 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":"2025-02-18","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 : 2025-02-13DOI: 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":"2025-02-13","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 : 2025-02-13DOI: 10.1016/j.jat.2025.106151
Deimer J.J. Aleans , Sergio A. Tozoni
<div><div>In this paper we investigate the asymptotic behavior of entropy numbers of multiplier operators <span><math><msub><mrow><mi>Λ</mi></mrow><mrow><mo>∗</mo></mrow></msub></math></span> and <span><math><mi>Λ</mi></math></span>, defined for functions on the complex sphere <span><math><msub><mrow><mi>Ω</mi></mrow><mrow><mi>d</mi></mrow></msub></math></span> of <span><math><msup><mrow><mi>ℂ</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>, associated with sequences of multipliers of the type <span><math><msub><mrow><mrow><mo>{</mo><msubsup><mrow><mi>λ</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>n</mi></mrow><mrow><mo>∗</mo></mrow></msubsup><mo>}</mo></mrow></mrow><mrow><mi>m</mi><mo>,</mo><mi>n</mi><mo>∈</mo><mi>N</mi></mrow></msub></math></span>, <span><math><mrow><msubsup><mrow><mi>λ</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>n</mi></mrow><mrow><mo>∗</mo></mrow></msubsup><mo>=</mo><mi>λ</mi><mrow><mo>(</mo><mi>m</mi><mo>+</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><msub><mrow><mrow><mo>{</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>n</mi></mrow></msub><mo>}</mo></mrow></mrow><mrow><mi>m</mi><mo>,</mo><mi>n</mi><mo>∈</mo><mi>N</mi></mrow></msub></math></span>, <span><math><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>n</mi></mrow></msub><mo>=</mo><mi>λ</mi><mrow><mo>(</mo><mo>max</mo><mrow><mo>{</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>}</mo></mrow><mo>)</mo></mrow></mrow></math></span>, respectively, for a bounded function <span><math><mi>λ</mi></math></span> defined on <span><math><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow></math></span>. If the operators <span><math><msub><mrow><mi>Λ</mi></mrow><mrow><mo>∗</mo></mrow></msub></math></span> and <span><math><mi>Λ</mi></math></span> are bounded from <span><math><mrow><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mrow><mo>(</mo><msub><mrow><mi>Ω</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span> into <span><math><mrow><msup><mrow><mi>L</mi></mrow><mrow><mi>q</mi></mrow></msup><mrow><mo>(</mo><msub><mrow><mi>Ω</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span>, <span><math><mrow><mn>1</mn><mo>≤</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>≤</mo><mi>∞</mi></mrow></math></span>, and <span><math><msub><mrow><mi>U</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> is the closed unit ball of <span><math><mrow><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mrow><mo>(</mo><msub><mrow><mi>Ω</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span>, we study lower and upper estimates for the entropy numbers of the sets <span><math><mrow><msub><mrow><mi>Λ</mi></mrow><mrow><mo>∗</mo></mrow></msub><msub><mrow><mi>U</mi></mrow><mrow><mi>p</mi></mrow></msub></mrow></math></span> and <span><math><mrow><mi>Λ</mi><msub><mrow><mi>U</mi></mrow><mrow><mi>p</mi></mrow></msub></mrow></math></span> in <span><math><mrow><msup><mrow><mi
{"title":"Estimates for entropy numbers of sets of smooth functions on complex spheres","authors":"Deimer J.J. Aleans , Sergio A. Tozoni","doi":"10.1016/j.jat.2025.106151","DOIUrl":"10.1016/j.jat.2025.106151","url":null,"abstract":"<div><div>In this paper we investigate the asymptotic behavior of entropy numbers of multiplier operators <span><math><msub><mrow><mi>Λ</mi></mrow><mrow><mo>∗</mo></mrow></msub></math></span> and <span><math><mi>Λ</mi></math></span>, defined for functions on the complex sphere <span><math><msub><mrow><mi>Ω</mi></mrow><mrow><mi>d</mi></mrow></msub></math></span> of <span><math><msup><mrow><mi>ℂ</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>, associated with sequences of multipliers of the type <span><math><msub><mrow><mrow><mo>{</mo><msubsup><mrow><mi>λ</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>n</mi></mrow><mrow><mo>∗</mo></mrow></msubsup><mo>}</mo></mrow></mrow><mrow><mi>m</mi><mo>,</mo><mi>n</mi><mo>∈</mo><mi>N</mi></mrow></msub></math></span>, <span><math><mrow><msubsup><mrow><mi>λ</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>n</mi></mrow><mrow><mo>∗</mo></mrow></msubsup><mo>=</mo><mi>λ</mi><mrow><mo>(</mo><mi>m</mi><mo>+</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><msub><mrow><mrow><mo>{</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>n</mi></mrow></msub><mo>}</mo></mrow></mrow><mrow><mi>m</mi><mo>,</mo><mi>n</mi><mo>∈</mo><mi>N</mi></mrow></msub></math></span>, <span><math><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>n</mi></mrow></msub><mo>=</mo><mi>λ</mi><mrow><mo>(</mo><mo>max</mo><mrow><mo>{</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>}</mo></mrow><mo>)</mo></mrow></mrow></math></span>, respectively, for a bounded function <span><math><mi>λ</mi></math></span> defined on <span><math><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow></math></span>. If the operators <span><math><msub><mrow><mi>Λ</mi></mrow><mrow><mo>∗</mo></mrow></msub></math></span> and <span><math><mi>Λ</mi></math></span> are bounded from <span><math><mrow><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mrow><mo>(</mo><msub><mrow><mi>Ω</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span> into <span><math><mrow><msup><mrow><mi>L</mi></mrow><mrow><mi>q</mi></mrow></msup><mrow><mo>(</mo><msub><mrow><mi>Ω</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span>, <span><math><mrow><mn>1</mn><mo>≤</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>≤</mo><mi>∞</mi></mrow></math></span>, and <span><math><msub><mrow><mi>U</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> is the closed unit ball of <span><math><mrow><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mrow><mo>(</mo><msub><mrow><mi>Ω</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span>, we study lower and upper estimates for the entropy numbers of the sets <span><math><mrow><msub><mrow><mi>Λ</mi></mrow><mrow><mo>∗</mo></mrow></msub><msub><mrow><mi>U</mi></mrow><mrow><mi>p</mi></mrow></msub></mrow></math></span> and <span><math><mrow><mi>Λ</mi><msub><mrow><mi>U</mi></mrow><mrow><mi>p</mi></mrow></msub></mrow></math></span> in <span><math><mrow><msup><mrow><mi","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"308 ","pages":"Article 106151"},"PeriodicalIF":0.9,"publicationDate":"2025-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143420993","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 : 2025-02-11DOI: 10.1016/j.jat.2025.106149
Jacek Gulgowski , Anna Kamont , Markus Passenbrunner
Let be a probability space and let be a binary filtration. i.e. exactly one atom of is divided into two atoms of without any restriction on their respective measures. Additionally, denote the collection of atoms corresponding to this filtration by . Let be a finite-dimensional linear subspace, having an additional stability property on atoms . For these data, we consider two dictionaries:
•
,
•
– a local orthonormal system generated by and the filtration .
Let , with . We are interested in approximation spaces and
设（Ω，，P）是一个概率空间，设(n)n=0∞是一个二元滤波。即，精确地将一个原子（n−1）分成两个原子（n−1）而不限制它们各自的度量。另外，用a表示与此过滤对应的原子集合。设S∧L∞（Ω）是一个有限维线性子空间，在原子a上具有额外的稳定性。对于这些数据，我们考虑两个字典：•C={f⋅1A:f∈S, a∈a}，•Φ—一个由S生成的局部正交系统和过滤(n)n=0∞。让Lp (S) =¯Lp(Ω)C =跨¯Lp(Ω)Φ1 & lt;术中;∞。我们对近似空间Aqα（Lp(S)，C）和Aqα（Lp(S)，Φ）感兴趣，它们分别对应于C和Φ的元素在Lp(S)中的最佳n项近似，其中α>；0和0<；q≤∞。已知在经典Haar情况下，即当S=span（1[0,1]）且二元滤除(n)n=0∞是二进的（即原子A∈A被分成两个新的等测度原子），我们有Aqα（Lp(S)，Φ）=Aqα(Lp(S)，C), cf. P. Petrushev（2003）。这促使我们问这样一个问题：在上述的一般情况下，这个等式是否成立？这个问题的答案取决于一个特定的Bernstein型不等式BI（a，S,p,τ）的有效性，其参数为1<；p<∞,0<τ<p。本文的主要结果是该类Bernstein不等式BI（a，S,p,τ）的一个几何刻划，即关于原子a和环上空间S上的函数的行为的一个刻划={a∈B: a,B∈a,B∧a}} a。我们将这个一般结果专门用于一些感兴趣的例子，包括一般Haar系统和由（多元）多项式组成的空间S。
{"title":"Properties of local orthonormal systems, Part II: Geometric characterization of Bernstein inequalities","authors":"Jacek Gulgowski , Anna Kamont , Markus Passenbrunner","doi":"10.1016/j.jat.2025.106149","DOIUrl":"10.1016/j.jat.2025.106149","url":null,"abstract":"<div><div>Let <span><math><mrow><mo>(</mo><mi>Ω</mi><mo>,</mo><mi>ℱ</mi><mo>,</mo><mi>P</mi><mo>)</mo></mrow></math></span> be a probability space and let <span><math><msubsup><mrow><mrow><mo>(</mo><msub><mrow><mi>ℱ</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> be a binary filtration. i.e. exactly one atom of <span><math><msub><mrow><mi>ℱ</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></math></span> is divided into <em>two</em> atoms of <span><math><msub><mrow><mi>ℱ</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> without any restriction on their respective measures. Additionally, denote the collection of atoms corresponding to this filtration by <span><math><mi>A</mi></math></span>. Let <span><math><mrow><mi>S</mi><mo>⊂</mo><msup><mrow><mi>L</mi></mrow><mrow><mi>∞</mi></mrow></msup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow></mrow></math></span> be a finite-dimensional linear subspace, having an additional stability property on atoms <span><math><mi>A</mi></math></span>. For these data, we consider two dictionaries: <ul><li><span>•</span><span><div><span><math><mrow><mi>C</mi><mo>=</mo><mrow><mo>{</mo><mi>f</mi><mi>⋅</mi><msub><mrow><mi>1</mi></mrow><mrow><mi>A</mi></mrow></msub><mo>:</mo><mi>f</mi><mo>∈</mo><mi>S</mi><mo>,</mo><mi>A</mi><mo>∈</mo><mi>A</mi><mo>}</mo></mrow></mrow></math></span>,</div></span></li><li><span>•</span><span><div><span><math><mi>Φ</mi></math></span> – a local orthonormal system generated by <span><math><mi>S</mi></math></span> and the filtration <span><math><msubsup><mrow><mrow><mo>(</mo><msub><mrow><mi>ℱ</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>.</div></span></li></ul></div><div>Let <span><math><mrow><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mrow><mo>(</mo><mi>S</mi><mo>)</mo></mrow><mo>=</mo><msub><mrow><mover><mrow><mi>span</mi></mrow><mo>¯</mo></mover></mrow><mrow><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow></mrow></msub><mi>C</mi><mo>=</mo><msub><mrow><mover><mrow><mi>span</mi></mrow><mo>¯</mo></mover></mrow><mrow><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow></mrow></msub><mi>Φ</mi></mrow></math></span>, with <span><math><mrow><mn>1</mn><mo><</mo><mi>p</mi><mo><</mo><mi>∞</mi></mrow></math></span>. We are interested in approximation spaces <span><math><mrow><msubsup><mrow><mi>A</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>α</mi></mrow></msubsup><mrow><mo>(</mo><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mrow><mo>(</mo><mi>S</mi><mo>)</mo></mrow><mo>,</mo><mi>C</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><msubsup><mrow><mi>A</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>α</mi></mrow></msubsup><mrow><mo>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"308 ","pages":"Article 106149"},"PeriodicalIF":0.9,"publicationDate":"2025-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143428911","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 : 2025-02-07DOI: 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":"2025-02-07","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}
Pub 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":"2025-02-06","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 : 2025-02-06DOI: 10.1016/j.jat.2025.106146
Magomed-Kasumov M.G
It is shown that for any absolutely continuous function on , the Fourier series with respect to the Jacobi polynomials converges uniformly on to this function if and only if , .
{"title":"Uniform convergence of Fourier–Jacobi series to absolutely continuous functions","authors":"Magomed-Kasumov M.G","doi":"10.1016/j.jat.2025.106146","DOIUrl":"10.1016/j.jat.2025.106146","url":null,"abstract":"<div><div>It is shown that for any absolutely continuous function on <span><math><mrow><mo>[</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></math></span>, the Fourier series with respect to the Jacobi polynomials <span><math><msubsup><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>α</mi><mo>,</mo><mi>β</mi></mrow></msubsup></math></span> converges uniformly on <span><math><mrow><mo>[</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></math></span> to this function if and only if <span><math><mrow><mo>−</mo><mn>1</mn><mo><</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>≤</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></math></span>, <span><math><mrow><mrow><mo>|</mo><mi>α</mi><mo>−</mo><mi>β</mi><mo>|</mo></mrow><mo>≤</mo><mn>1</mn></mrow></math></span>.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"308 ","pages":"Article 106146"},"PeriodicalIF":0.9,"publicationDate":"2025-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143379077","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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