{"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}
{"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}
Chelo Ferreira , José L. López , Ester Pérez Sinusía
{"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}
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