Journal of Approximation Theory最新文献

{"title":"Asymptotics of the Humbert functions Ψ1 and Ψ2","authors":"Peng-Cheng Hang ,&nbsp;Malte Henkel ,&nbsp;Min-Jie Luo","doi":"10.1016/j.jat.2025.106233","DOIUrl":"10.1016/j.jat.2025.106233","url":null,"abstract":"<div><div>A compilation of new results on the asymptotic behaviour of the Humbert functions <span><math><msub><mrow><mi>Ψ</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>Ψ</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>, and also on the Appell function <span><math><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>, is presented. As a by-product, we confirm a conjectured limit which appeared recently in the study of the <span><math><mrow><mn>1</mn><mi>D</mi></mrow></math></span> Glauber–Ising model. We also propose two elementary asymptotic methods and confirm through some illustrative examples that both methods have great potential and can be applied to a large class of problems of asymptotic analysis. Finally, some directions of future research are pointed out in order to suggest ideas for further study.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"314 ","pages":"Article 106233"},"PeriodicalIF":0.6,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145027835","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":"Widom factors in ℂn","authors":"Gökalp Alpan ,&nbsp;Turgay Bayraktar ,&nbsp;Norm Levenberg","doi":"10.1016/j.jat.2025.106227","DOIUrl":"10.1016/j.jat.2025.106227","url":null,"abstract":"<div><div>We generalize the theory of Widom factors to the <span><math><msup><mrow><mi>ℂ</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> setting. We define Widom factors of compact subsets <span><math><mrow><mi>K</mi><mo>⊂</mo><msup><mrow><mi>ℂ</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></math></span> associated with multivariate orthogonal polynomials and weighted Chebyshev polynomials. We show that on product subsets <span><math><mrow><mi>K</mi><mo>=</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>×</mo><mo>⋯</mo><mo>×</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></math></span> of <span><math><msup><mrow><mi>ℂ</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>, where each <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>j</mi></mrow></msub></math></span> is a non-polar compact subset of <span><math><mi>ℂ</mi></math></span>, these quantities have universal lower bounds which directly extend one dimensional results. Under the additional assumption that each <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>j</mi></mrow></msub></math></span> is a subset of the real line, we provide improved lower bounds for Widom factors for some weight functions <span><math><mi>w</mi></math></span>; in particular, for the case <span><math><mrow><mi>w</mi><mo>≡</mo><mn>1</mn></mrow></math></span>. Finally, we define the Mahler measure of a multivariate polynomial relative to <span><math><mrow><mi>K</mi><mo>⊂</mo><msup><mrow><mi>ℂ</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></math></span> and obtain lower bounds for this quantity on product sets.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"313 ","pages":"Article 106227"},"PeriodicalIF":0.6,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144887235","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":"On the extremal eigenvalues of Jacobi ensembles at zero temperature","authors":"Kilian Hermann,&nbsp;Michael Voit","doi":"10.1016/j.jat.2025.106229","DOIUrl":"10.1016/j.jat.2025.106229","url":null,"abstract":"<div><div>For the <span><math><mi>β</mi></math></span>-Hermite, Laguerre, and Jacobi ensembles of dimension <span><math><mi>N</mi></math></span> there exist central limit theorems for the freezing case <span><math><mrow><mi>β</mi><mo>→</mo><mi>∞</mi></mrow></math></span> such that the associated means and covariances can be expressed in terms of the associated Hermite, Laguerre, and Jacobi polynomials of order <span><math><mi>N</mi></math></span> respectively as well as via the associated dual polynomials in the sense of de Boor and Saff. In this paper we derive limits for <span><math><mrow><mi>N</mi><mo>→</mo><mi>∞</mi></mrow></math></span> for the covariances of the <span><math><mrow><mi>r</mi><mo>∈</mo><mi>N</mi></mrow></math></span> largest (and smallest) eigenvalues for these frozen Jacobi ensembles in terms of Bessel functions. These results correspond to the hard edge analysis in the frozen Laguerre cases by Andraus and Lerner-Brecher and to known results for finite <span><math><mi>β</mi></math></span>.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"313 ","pages":"Article 106229"},"PeriodicalIF":0.6,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145578824","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":"Sharp iteration asymptotics for transfer operators induced by greedy β-expansions","authors":"Horia D. Cornean,&nbsp;Kasper S. Sørensen","doi":"10.1016/j.jat.2025.106234","DOIUrl":"10.1016/j.jat.2025.106234","url":null,"abstract":"<div><div>We consider base-<span><math><mi>β</mi></math></span> expansions of Parry’s type, where <span><math><mrow><msub><mrow><mi>a</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>≥</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>≥</mo><mn>1</mn></mrow></math></span> are integers and <span><math><mrow><msub><mrow><mi>a</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>&lt;</mo><mi>β</mi><mo>&lt;</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>+</mo><mn>1</mn></mrow></math></span> is the positive solution to <span><math><mrow><msup><mrow><mi>β</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>0</mn></mrow></msub><mi>β</mi><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></math></span> (the golden ratio corresponds to <span><math><mrow><msub><mrow><mi>a</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>=</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mn>1</mn></mrow></math></span>). The map <span><math><mrow><mi>x</mi><mo>↦</mo><mi>β</mi><mi>x</mi><mo>−</mo><mrow><mo>⌊</mo><mi>β</mi><mi>x</mi><mo>⌋</mo></mrow></mrow></math></span> induces a discrete dynamical system on the interval <span><math><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></math></span> and we study its associated transfer (Perron–Frobenius) operator <span><math><mi>P</mi></math></span>. Our main result can be roughly summarized as follows: we explicitly construct two piecewise affine functions <span><math><mi>u</mi></math></span> and <span><math><mi>v</mi></math></span> with <span><math><mrow><mi>P</mi><mi>u</mi><mo>=</mo><mi>u</mi></mrow></math></span> and <span><math><mrow><mi>P</mi><mi>v</mi><mo>=</mo><msup><mrow><mi>β</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mi>v</mi></mrow></math></span> such that for every sufficiently smooth <span><math><mi>F</mi></math></span> which is supported in <span><math><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></math></span> and satisfies <span><math><mrow><msubsup><mrow><mo>∫</mo></mrow><mrow><mn>0</mn></mrow><mrow><mn>1</mn></mrow></msubsup><mi>F</mi><mspace></mspace><mi>d</mi><mi>x</mi><mo>=</mo><mn>1</mn></mrow></math></span>, we have <span><math><mrow><msup><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msup><mi>F</mi><mo>=</mo><mi>u</mi><mo>+</mo><msup><mrow><mi>β</mi></mrow><mrow><mo>−</mo><mi>k</mi></mrow></msup><mrow><mo>(</mo><mrow><mi>F</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow><mo>−</mo><mi>F</mi><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mi>v</mi><mo>+</mo><mi>o</mi><mrow><mo>(</mo><msup><mrow><mi>β</mi></mrow><mrow><mo>−</mo><mi>k</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span> in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>∞</mi></mrow></msup></math></span>. This is also compared with the case of integer bases, where more refined asymptotic formulas are possible.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"313 ","pages":"Article 106234"},"PeriodicalIF":0.6,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145026503","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":"On positive Jacobi matrices with compact inverses","authors":"Pavel Šťovíček ,&nbsp;Grzegorz Świderski","doi":"10.1016/j.jat.2025.106217","DOIUrl":"10.1016/j.jat.2025.106217","url":null,"abstract":"<div><div>We consider positive Jacobi matrices <span><math><mi>J</mi></math></span> with compact inverses and consequently with purely discrete spectra. A number of properties of the corresponding sequence of orthogonal polynomials is studied including the convergence of their zeros, the vague convergence of the zero counting measures and of the Christoffel–Darboux kernels on the diagonal. Particularly, if the inverse of <span><math><mi>J</mi></math></span> belongs to some Schatten class, we identify the asymptotic behavior of the sequence of orthogonal polynomials and express it in terms of its regularized characteristic function. In the even more special case when the inverse of <span><math><mi>J</mi></math></span> belongs to the trace class, we derive various formulas for the orthogonality measure, eigenvectors of <span><math><mi>J</mi></math></span> as well as for the functions of the second kind and related objects. These general results are given a more explicit form in the case when <span><math><mrow><mo>−</mo><mi>J</mi></mrow></math></span> is a generator of a Birth–Death process. Among others, we provide a formula for the trace of the inverse of <span><math><mi>J</mi></math></span>. We illustrate our results by introducing and studying a modification of the <span><math><mi>q</mi></math></span>-Laguerre polynomials corresponding to a determinate moment problem.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"313 ","pages":"Article 106217"},"PeriodicalIF":0.6,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144763703","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":"In Memoriam: Konstantin Il’ich Oskolkov February 17, 1946–August 24, 2024","authors":"Boris Kashin,&nbsp;Alexander Stokolos,&nbsp;Vladimir Temlyakov","doi":"10.1016/j.jat.2025.106243","DOIUrl":"10.1016/j.jat.2025.106243","url":null,"abstract":"","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"313 ","pages":"Article 106243"},"PeriodicalIF":0.6,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145578826","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":"A 3 × 3 singular solution to the Matrix Bochner Problem with D(W) not of the form ℂ[D]","authors":"Ignacio Bono Parisi","doi":"10.1016/j.jat.2025.106247","DOIUrl":"10.1016/j.jat.2025.106247","url":null,"abstract":"<div><div>The Matrix Bochner Problem aims to classify weight matrices whose sequences of orthogonal polynomials are eigenfunctions of a second-order differential operator. A major breakthrough in this direction was achieved in Casper and Yakimov (2022), where it was shown that, under certain natural conditions on the algebra <span><math><mrow><mi>D</mi><mrow><mo>(</mo><mi>W</mi><mo>)</mo></mrow></mrow></math></span>, all solutions arise from Darboux transformations of direct sums of classical scalar weights. In this paper, we study a new 3 × 3 Hermite-type weight matrix and determine its algebra <span><math><mrow><mi>D</mi><mrow><mo>(</mo><mi>W</mi><mo>)</mo></mrow></mrow></math></span> as a <span><math><mrow><mi>ℂ</mi><mrow><mo>[</mo><msub><mrow><mi>D</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>]</mo></mrow></mrow></math></span>-module generated by <span><math><mrow><mo>{</mo><mi>I</mi><mo>,</mo><msub><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>}</mo></mrow></math></span>, where <span><math><msub><mrow><mi>D</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> are second-order differential operators. This complete description of the algebra allows us to prove that the weight does not arise from a Darboux transformation of classical scalar weights, showing that it falls outside the classification theorem of Casper and Yakimov (2022). Unlike previous examples in Bono Parisi and Pacharoni (2024) [3, 4], which also do not fit within this classification, the algebra <span><math><mrow><mi>D</mi><mrow><mo>(</mo><mi>W</mi><mo>)</mo></mrow></mrow></math></span> of this weight matrix is not generated by a single differential operator <span><math><mi>D</mi></math></span>, making it a fundamentally different case. These results complement the classification theorem of the Matrix Bochner Problem by providing a new type of singular example.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"313 ","pages":"Article 106247"},"PeriodicalIF":0.6,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145465905","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":"Sharp lower bounds on interpolation by deep ReLU neural networks at irregularly spaced data","authors":"Jonathan W. Siegel","doi":"10.1016/j.jat.2025.106244","DOIUrl":"10.1016/j.jat.2025.106244","url":null,"abstract":"<div><div>We study the interpolation power of deep ReLU neural networks. Specifically, we consider the question of how efficiently, in terms of the number of parameters, deep ReLU networks can interpolate values at <span><math><mi>N</mi></math></span> datapoints in the unit ball which are separated by a distance <span><math><mi>δ</mi></math></span>. We show that <span><math><mrow><mi>Ω</mi><mrow><mo>(</mo><mi>N</mi><mo>)</mo></mrow></mrow></math></span> parameters are required in the regime where <span><math><mi>δ</mi></math></span> is exponentially small in <span><math><mi>N</mi></math></span>, which gives the sharp result in this regime since <span><math><mrow><mi>O</mi><mrow><mo>(</mo><mi>N</mi><mo>)</mo></mrow></mrow></math></span> parameters are always sufficient. This also shows that the bit-extraction technique used to prove lower bounds on the VC dimension cannot be applied to irregularly spaced datapoints. Finally, as an application we give a lower bound on the approximation rates that deep ReLU neural networks can achieve for Sobolev spaces at the embedding endpoint.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"313 ","pages":"Article 106244"},"PeriodicalIF":0.6,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145362068","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":"A characterization of completely alternating functions","authors":"Monojit Bhattacharjee ,&nbsp;Rajkamal Nailwal","doi":"10.1016/j.jat.2025.106230","DOIUrl":"10.1016/j.jat.2025.106230","url":null,"abstract":"<div><div>In this article, we characterize completely alternating functions on an abelian semigroup <span><math><mi>S</mi></math></span> in terms of completely monotone functions on the product semigroup <span><math><mrow><mi>S</mi><mo>×</mo><msub><mrow><mi>Z</mi></mrow><mrow><mo>+</mo></mrow></msub></mrow></math></span>. We also discuss completely alternating sequences induced by a class of rational functions and obtain a set of sufficient conditions (in terms of its zeros and poles) to determine them. As an application, we show a complete characterization of several classes of completely monotone functions on <span><math><msubsup><mrow><mi>Z</mi></mrow><mrow><mo>+</mo></mrow><mrow><mn>2</mn></mrow></msubsup></math></span> induced by rational functions in two variables. We also derive a set of necessary conditions for the complete monotonicity of the sequence <span><math><mrow><msub><mrow><mrow><mo>{</mo><msubsup><mrow><mo>∏</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>k</mi></mrow></msubsup><mfrac><mrow><mrow><mo>(</mo><mi>n</mi><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo></mrow></mrow><mrow><mrow><mo>(</mo><mi>n</mi><mo>+</mo><msub><mrow><mi>b</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo></mrow></mrow></mfrac><mo>}</mo></mrow></mrow><mrow><mi>n</mi><mo>∈</mo><msub><mrow><mi>Z</mi></mrow><mrow><mo>+</mo></mrow></msub></mrow></msub><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>b</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∈</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow></mrow></math></span>.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"313 ","pages":"Article 106230"},"PeriodicalIF":0.6,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144890100","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":"Simplified uniform asymptotic expansions for associated Legendre and conical functions","authors":"T.M. Dunster","doi":"10.1016/j.jat.2025.106228","DOIUrl":"10.1016/j.jat.2025.106228","url":null,"abstract":"<div><div>Asymptotic expansions are derived for associated Legendre functions of degree <span><math><mi>ν</mi></math></span> and order <span><math><mi>μ</mi></math></span>, where one or the other of the parameters is large. The expansions are uniformly valid for unbounded real and complex values of the argument <span><math><mi>z</mi></math></span>, including the singularity <span><math><mrow><mi>z</mi><mo>=</mo><mn>1</mn></mrow></math></span>. The cases where <span><math><mrow><mi>ν</mi><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></math></span> and <span><math><mi>μ</mi></math></span> are real or purely imaginary are included, which includes conical functions. The approximations involve either exponential or modified Bessel functions, along with slowly varying coefficient functions. The coefficients of the new asymptotic expansions are simple and readily obtained explicitly, allowing for computation to a high degree of accuracy. The results are constructed and rigorously established by employing certain Liouville–Green type expansions where the coefficients appear in the exponent of an exponential function.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"313 ","pages":"Article 106228"},"PeriodicalIF":0.6,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145578827","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}