Journal of Approximation Theory最新文献

{"title":"In Memoriam: Konstantin Il’ich Oskolkov February 17, 1946–August 24, 2024","authors":"Boris Kashin, Alexander Stokolos, 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":"2025-10-23","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":"2-strong uniqueness of a best approximation and of minimal projections in complex polytope norms and their duals","authors":"Tomasz Kobos,&nbsp;Grzegorz Lewicki","doi":"10.1016/j.jat.2025.106245","DOIUrl":"10.1016/j.jat.2025.106245","url":null,"abstract":"<div><div>We study a property of 2-strong uniqueness of a best approximation in a class of finite-dimensional complex normed spaces, for which the unit ball is an absolutely convex hull of finite number of points and in its dual class. We prove that, contrary to the real case, these two classes do not coincide but are in fact disjoint. We provide several examples of situations in these two classes, where a uniqueness of an element of a best approximation in a given subspace implies its 2-strong uniqueness. In particular, such a property holds for approximation in an arbitrary subspace of the complex <span><math><msubsup><mrow><mi>ℓ</mi></mrow><mrow><mn>1</mn></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> space, but not of the complex <span><math><msubsup><mrow><mi>ℓ</mi></mrow><mrow><mi>∞</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> space. However, this is true in general under an additional assumption that a subspace has a real basis and an ambient complex normed space is generated by real vectors or functionals. We apply our results and related methods to establish some results concerned with 2-strongly unique minimal projections in complex normed spaces, proving among other things, that a minimal projection onto a two-dimensional subspace of an arbitrary three-dimensional complex normed space is 2-strongly unique, if its norm is greater than 1.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"314 ","pages":"Article 106245"},"PeriodicalIF":0.6,"publicationDate":"2025-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145363620","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":"2025-10-15","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":"Nonlinear approximation of harmonic functions from shifts of the Newtonian kernel in BMO","authors":"Kamen G. Ivanov ,&nbsp;Pencho Petrushev","doi":"10.1016/j.jat.2025.106246","DOIUrl":"10.1016/j.jat.2025.106246","url":null,"abstract":"<div><div>We study nonlinear <span><math><mi>n</mi></math></span>-term approximation of harmonic functions on the unit ball in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> from linear combinations of shifts of the Newtonian kernel (fundamental solution of the Laplace equation) in BMO. A Jackson estimate is established that naturally involves Besov spaces lying on the Sobolev embedding line for BMO. The method for obtaining this result is based on the construction of highly localized frames for Besov spaces and VMO on the sphere whose elements are linear combinations of a fixed number of shifts of the Newtonian kernel.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"314 ","pages":"Article 106246"},"PeriodicalIF":0.6,"publicationDate":"2025-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145325809","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":"2025-09-03","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":"Nevai’s condition for measures with unbounded supports","authors":"Grzegorz Świderski","doi":"10.1016/j.jat.2025.106232","DOIUrl":"10.1016/j.jat.2025.106232","url":null,"abstract":"<div><div>We study Nevai’s condition from the theory of orthogonal polynomials on the real line. We prove that a large class of measures with unbounded Jacobi parameters satisfies Nevai’s condition locally uniformly on the support of the measure away from a finite explicit set. This allows us to give applications to relative uniform and weak asymptotics of Christoffel–Darboux kernels on the diagonal and to limit theorems for unconventionally normalized global linear statistics of orthogonal polynomial ensembles.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"314 ","pages":"Article 106232"},"PeriodicalIF":0.6,"publicationDate":"2025-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145050628","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 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":"2025-09-03","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":"Interpolation of compact multilinear operators between quasi-Banach spaces","authors":"Fernando Cobos ,&nbsp;Luz M. Fernández-Cabrera ,&nbsp;Thomas Kühn","doi":"10.1016/j.jat.2025.106222","DOIUrl":"10.1016/j.jat.2025.106222","url":null,"abstract":"<div><div>We investigate the interpolation properties of compact multilinear operators by the real method between quasi-Banach spaces. As an application we establish a reinforced version of a multilinear Marcinkiewicz theorem.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"314 ","pages":"Article 106222"},"PeriodicalIF":0.6,"publicationDate":"2025-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145683675","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":"Dedication","authors":"Alexander Brudnyi,&nbsp;Natan Kruglyak,&nbsp;Mieczysław Mastyło,&nbsp;Paul Nevai,&nbsp;Amos Ron,&nbsp;Pavel Shvartsman","doi":"10.1016/j.jat.2025.106231","DOIUrl":"10.1016/j.jat.2025.106231","url":null,"abstract":"","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"314 ","pages":"Article 106231"},"PeriodicalIF":0.6,"publicationDate":"2025-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145683678","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":"Invertible and Fredholm operators on interpolation scales","authors":"Irina Asekritova ,&nbsp;Natan Kruglyak ,&nbsp;Mieczysław Mastyło","doi":"10.1016/j.jat.2025.106213","DOIUrl":"10.1016/j.jat.2025.106213","url":null,"abstract":"<div><div>We investigate the behaviour of invertible and Fredholm operators on interpolation scales constructed via a family of interpolation functors <span><math><msub><mrow><mrow><mo>{</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>θ</mi></mrow></msub><mo>}</mo></mrow></mrow><mrow><mi>θ</mi><mo>∈</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></mrow></msub></math></span>. This family includes both complex and real interpolation functors. Our results demonstrate, in particular, that kernels and cokernels of operators are stable on intervals of parameters <span><math><mi>θ</mi></math></span> where the operators are Fredholm. Additionally, we introduce the notion of Fredholm operators in the category of Banach couples, establishing its relevance for the obtained results.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"314 ","pages":"Article 106213"},"PeriodicalIF":0.6,"publicationDate":"2025-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145683676","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}