Bulletin des Sciences Mathematiques最新文献

{"title":"On near orthogonality of the Banach frames for the wave packet spaces","authors":"Dimitri Bytchenkoff","doi":"10.1016/j.bulsci.2025.103611","DOIUrl":"10.1016/j.bulsci.2025.103611","url":null,"abstract":"<div><div>In solving scientific, engineering or pure mathematical problems one is frequently faced with a need to approximate the function of a given class with a specified precision by the linear combination of a preferably small number of simpler functions. This can often achieved by choosing the simpler functions localised one way or another both in the time and frequency domain. Constructing a set of linearly independent functions, let alone a basis, with a given time-frequency localisation is a formidable and often unsolvable problem, though. A much better chance one stands in building a set of time-frequency localised functions that constitutes a so-called frame – a generalisation of the notion of the basis, whose elements need not be linear independent, rather than a basis.</div><div>Over the last seventy years or so, a range of frames have been designed to allow the decomposition and synthesis of functions of various classes. The most prominent examples of such systems are Gabor functions, wavelets, ridgelets, curvelets, shearlets and wave atoms. We recently introduced a family of quasi-Banach spaces – which we called <span><math><mi>w</mi><mi>a</mi><mi>v</mi><mi>e</mi></math></span> <span><math><mi>p</mi><mi>a</mi><mi>c</mi><mi>k</mi><mi>e</mi><mi>t</mi></math></span> <span><math><mi>s</mi><mi>p</mi><mi>a</mi><mi>c</mi><mi>e</mi><mi>s</mi></math></span> – that encompasses all those classes of functions whose elements have sparse expansions in one of the above-mentioned frames, supplied them with Banach frames – the kind of frames that ensure that any element of the class of functions for which a frame was designed can be decomposed and reconstructed using that frame – and provided their atomic decomposition. Herein we prove that the Banach frames for and sets of atoms of the wave packet spaces – which we call <span><math><mi>w</mi><mi>a</mi><mi>v</mi><mi>e</mi></math></span> <span><math><mi>p</mi><mi>a</mi><mi>c</mi><mi>k</mi><mi>e</mi><mi>t</mi></math></span> <span><math><mi>s</mi><mi>y</mi><mi>s</mi><mi>t</mi><mi>e</mi><mi>m</mi><mi>s</mi></math></span> – are indeed localised in the time and frequency domain or, more specifically, that they are near orthogonal; and therefore so are all of the above-mentioned examples of frames.</div><div>We shall also show that, unlike those examples, the wave packet system can be made to assume a wide range of types and degrees of time-frequency localisation by the suitable choice of values of the parameters of the system. This, we believe, makes the wave packet systems not only suitable for decomposing, synthesising or approximating functions of a wide range of quasi-Banach function spaces in an efficient and effective way, but also for their use for representing linear bounded operators on the quasi-Banach spaces by sparse and well structured matrices using the Galerkin method. This, in its turn, should allow one to design efficient computer programs for solving corresponding operator equations on t","PeriodicalId":55313,"journal":{"name":"Bulletin des Sciences Mathematiques","volume":"201 ","pages":"Article 103611"},"PeriodicalIF":1.3,"publicationDate":"2025-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143643317","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 estimates for Hausdorff operators on complementary local Morrey-type spaces","authors":"Mingquan Wei ,&nbsp;Xiaoyu Liu ,&nbsp;Dunyan Yan","doi":"10.1016/j.bulsci.2025.103614","DOIUrl":"10.1016/j.bulsci.2025.103614","url":null,"abstract":"<div><div>In this paper, some necessary and sufficient conditions for the boundedness of some linear and multilinear Hausdorff operators are established and the corresponding sharp constants are also given. Our main results can apply to some concrete operators, such as the Hardy operator and its adjoint operator, the weighted Hardy–Littlewood average, the multilinear Hardy operator and so on.</div></div>","PeriodicalId":55313,"journal":{"name":"Bulletin des Sciences Mathematiques","volume":"201 ","pages":"Article 103614"},"PeriodicalIF":1.3,"publicationDate":"2025-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143682116","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":"Inequalities for eigenvalues of the poly-Laplacian with arbitrary order on spherical domains","authors":"Yue He,&nbsp;Huan Wang","doi":"10.1016/j.bulsci.2025.103608","DOIUrl":"10.1016/j.bulsci.2025.103608","url":null,"abstract":"<div><div>In this paper, we are devoted to the study of universal inequalities for eigenvalues of the poly-Laplacian with arbitrary order on bounded domains in <span><math><msup><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>(</mo><mn>1</mn><mo>)</mo></math></span>, respectively, and then establish some new universal inequalities that are different from those already present in the literature. In particular, our results can reveal the relationship between the <span><math><mo>(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span>-th eigenvalue and the first <em>k</em> eigenvalues relatively quickly, and some methods used in this paper might be applied to other eigenvalue problems.</div></div>","PeriodicalId":55313,"journal":{"name":"Bulletin des Sciences Mathematiques","volume":"201 ","pages":"Article 103608"},"PeriodicalIF":1.3,"publicationDate":"2025-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143600544","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":"Weak weak approximation for certain quadric surface bundles","authors":"Nick Rome","doi":"10.1016/j.bulsci.2025.103601","DOIUrl":"10.1016/j.bulsci.2025.103601","url":null,"abstract":"<div><div>We investigate weak approximation away from a finite set of places for a class of biquadratic fourfolds inside <span><math><msup><mrow><mi>P</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>×</mo><msup><mrow><mi>P</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>, some of which appear in the recent work of Hassett–Pirutka–Tschinkel <span><span>[16]</span></span>.</div></div>","PeriodicalId":55313,"journal":{"name":"Bulletin des Sciences Mathematiques","volume":"201 ","pages":"Article 103601"},"PeriodicalIF":1.3,"publicationDate":"2025-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143551728","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":"On the essential norms of Toeplitz operators with symbols in C + H∞ acting on abstract Hardy spaces built upon translation-invariant Banach function spaces","authors":"Oleksiy Karlovych ,&nbsp;Eugene Shargorodsky","doi":"10.1016/j.bulsci.2025.103599","DOIUrl":"10.1016/j.bulsci.2025.103599","url":null,"abstract":"<div><div>Let <em>X</em> be a translation-invariant Banach function space on the unit circle and let <span><math><mi>H</mi><mo>[</mo><mi>X</mi><mo>]</mo></math></span> be the abstract Hardy space built upon <em>X</em>. We suppose the Riesz projection <em>P</em> is bounded on <em>X</em> and estimate the essential norms <span><math><msub><mrow><mo>‖</mo><mi>T</mi><mo>(</mo><mi>a</mi><mo>)</mo><mo>‖</mo></mrow><mrow><mi>B</mi><mo>(</mo><mi>H</mi><mo>[</mo><mi>X</mi><mo>]</mo><mo>)</mo><mo>,</mo><mi>e</mi></mrow></msub></math></span> of Toeplitz operators <span><math><mi>T</mi><mo>(</mo><mi>a</mi><mo>)</mo><mi>f</mi><mo>:</mo><mo>=</mo><mi>P</mi><mo>(</mo><mi>a</mi><mi>f</mi><mo>)</mo></math></span> with <span><math><mi>a</mi><mo>∈</mo><mi>C</mi><mo>+</mo><msup><mrow><mi>H</mi></mrow><mrow><mo>∞</mo></mrow></msup></math></span>. We prove that in this case<span><span><span><math><msub><mrow><mo>‖</mo><mi>a</mi><mo>‖</mo></mrow><mrow><msup><mrow><mi>L</mi></mrow><mrow><mo>∞</mo></mrow></msup></mrow></msub><mo>≤</mo><msub><mrow><mo>‖</mo><mi>T</mi><mo>(</mo><mi>a</mi><mo>)</mo><mo>‖</mo></mrow><mrow><mi>B</mi><mo>(</mo><mi>H</mi><mo>[</mo><mi>X</mi><mo>]</mo><mo>)</mo><mo>,</mo><mi>e</mi></mrow></msub><mo>≤</mo><mi>min</mi><mo>⁡</mo><mrow><mo>{</mo><mn>2</mn><mo>,</mo><msub><mrow><mo>‖</mo><mi>P</mi><mo>‖</mo></mrow><mrow><mi>B</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow></msub><mo>}</mo></mrow><msub><mrow><mo>‖</mo><mi>a</mi><mo>‖</mo></mrow><mrow><msup><mrow><mi>L</mi></mrow><mrow><mo>∞</mo></mrow></msup></mrow></msub><mo>,</mo></math></span></span></span> extending the results by the second author <span><span>[27]</span></span> for classical Hardy spaces <span><math><msup><mrow><mi>H</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>=</mo><mi>H</mi><mo>[</mo><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>]</mo></math></span>, <span><math><mn>1</mn><mo>&lt;</mo><mi>p</mi><mo>&lt;</mo><mo>∞</mo></math></span>. In contrast to our previous works <span><span>[27]</span></span> and <span><span>[16]</span></span>, we do not assume that <em>X</em> is reflexive or separable, which complicates the matters, but allows us to include the Hardy-Lorentz spaces <span><math><msup><mrow><mi>H</mi></mrow><mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msup><mo>=</mo><mi>H</mi><mo>[</mo><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msup><mo>]</mo></math></span> with <span><math><mn>1</mn><mo>&lt;</mo><mi>p</mi><mo>&lt;</mo><mo>∞</mo></math></span> and <span><math><mi>q</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>∞</mo></math></span> into consideration.</div></div>","PeriodicalId":55313,"journal":{"name":"Bulletin des Sciences Mathematiques","volume":"201 ","pages":"Article 103599"},"PeriodicalIF":1.3,"publicationDate":"2025-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143551727","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":"Schwarz lemma and Schwarz-Pick lemma for solutions of the α-harmonic equation","authors":"Ming Li ,&nbsp;Xiu-Shuang Ma ,&nbsp;Li-Mei Wang","doi":"10.1016/j.bulsci.2025.103598","DOIUrl":"10.1016/j.bulsci.2025.103598","url":null,"abstract":"<div><div>In this paper, the Schwarz type and Schwarz-Pick type inequalities for solutions of <em>α</em>-harmonic equation (<span><math><mi>α</mi><mo>&gt;</mo><mo>−</mo><mn>1</mn></math></span>) are investigated. By making use of the integral of trigonometric functions, we obtain these two types of inequalities in terms of hypergeometric functions which improve the corresponding results due to Khalfallah et al. (Complex Var. Elliptic Equ., 2023) and Li et al. (Bull. Malays. Math. Sci. Soc., 2022).</div></div>","PeriodicalId":55313,"journal":{"name":"Bulletin des Sciences Mathematiques","volume":"201 ","pages":"Article 103598"},"PeriodicalIF":1.3,"publicationDate":"2025-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143551791","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 representation of large even integers as the sum of eight primes from positive density sets","authors":"Meng Gao","doi":"10.1016/j.bulsci.2025.103597","DOIUrl":"10.1016/j.bulsci.2025.103597","url":null,"abstract":"<div><div>Let <span><math><mi>P</mi></math></span> denote the set of all primes. We have proved that if <em>A</em> is a subset of <span><math><mi>P</mi></math></span>, and the lower density of <em>A</em> in <span><math><mi>P</mi></math></span> is larger than 1/2, then every sufficiently large even integer <em>n</em> can be expressed in the form <span><math><mi>n</mi><mo>=</mo><msub><mrow><mi>p</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>p</mi></mrow><mrow><mn>8</mn></mrow></msub></math></span>, where <span><math><msub><mrow><mi>p</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>p</mi></mrow><mrow><mn>8</mn></mrow></msub><mo>∈</mo><mi>A</mi></math></span>. The constant 1/2 in this statement is the best possible.</div></div>","PeriodicalId":55313,"journal":{"name":"Bulletin des Sciences Mathematiques","volume":"201 ","pages":"Article 103597"},"PeriodicalIF":1.3,"publicationDate":"2025-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143510382","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":"Formal integrability for monodromic nilpotent singular points in R3","authors":"Claudio Pessoa,&nbsp;Lucas Queiroz","doi":"10.1016/j.bulsci.2025.103588","DOIUrl":"10.1016/j.bulsci.2025.103588","url":null,"abstract":"<div><div>Consider analytic three-dimensional differential systems having a singular point at the origin such that its linear part is <span><math><mi>y</mi><msub><mrow><mo>∂</mo></mrow><mrow><mi>x</mi></mrow></msub><mo>−</mo><mi>λ</mi><mi>z</mi><msub><mrow><mo>∂</mo></mrow><mrow><mi>z</mi></mrow></msub></math></span> for some <span><math><mi>λ</mi><mo>≠</mo><mn>0</mn></math></span>. The restriction of such systems to a center manifold has a nilpotent singular point at the origin. We study the formal and analytic integrability for those types of singular points in the monodromic case. As a byproduct, we obtain some useful results for planar <span><math><msup><mrow><mi>C</mi></mrow><mrow><mi>r</mi></mrow></msup></math></span> systems having a monodromic nilpotent singularity. We conclude the work by studying issues related to monodromy and formal integrability for the Elsonbaty–El-Sayed system, the Hide–Skeldon–Acheson dynamo system and the Generalized Lorenz system. For this last system, we were able to detect nilpotent centers.</div></div>","PeriodicalId":55313,"journal":{"name":"Bulletin des Sciences Mathematiques","volume":"200 ","pages":"Article 103588"},"PeriodicalIF":1.3,"publicationDate":"2025-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143509436","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":"Direct sums and abstract Kadets–Klee properties","authors":"Tomasz Kiwerski,&nbsp;Paweł Kolwicz","doi":"10.1016/j.bulsci.2025.103587","DOIUrl":"10.1016/j.bulsci.2025.103587","url":null,"abstract":"<div><div>Let <span><math><mi>X</mi><mo>=</mo><msub><mrow><mo>{</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>γ</mi></mrow></msub><mo>}</mo></mrow><mrow><mi>γ</mi><mo>∈</mo><mi>Γ</mi></mrow></msub></math></span> be a family of Banach spaces and let <span><math><mi>E</mi></math></span> be a Banach sequence space defined on Γ. The main aim of this work is to investigate the abstract Kadets–Klee properties, that is, the Kadets–Klee type properties in which the weak convergence of sequences is replaced by the convergence with respect to some linear Hausdorff topology, for the direct sum construction <span><math><msub><mrow><mo>(</mo><msub><mrow><mo>⨁</mo></mrow><mrow><mi>γ</mi><mo>∈</mo><mi>Γ</mi></mrow></msub><msub><mrow><mi>X</mi></mrow><mrow><mi>γ</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>E</mi></mrow></msub></math></span>. As we will show, and this seems to be quite atypical behavior when compared to some other geometric properties, to lift the Kadets–Klee properties from the components to whole direct sum it is not enough to assume that all involved spaces have the appropriate Kadets–Klee property. Actually, to complete the picture one must add a dichotomy in the form of the Schur type properties for <span><math><msub><mrow><mi>X</mi></mrow><mrow><mi>γ</mi></mrow></msub></math></span>'s supplemented by the variant of strict monotonicity for <span><math><mi>E</mi></math></span>. Back down to earth, this general machinery naturally provides a blue print for other topologies like, for example, the weak topology or the topology of local convergence in measure, that are perhaps more commonly associated with this type of considerations. Furthermore, by limiting ourselves to direct sums in which the family <span><math><mi>X</mi></math></span> is constant, that is, <span><math><msub><mrow><mi>X</mi></mrow><mrow><mi>γ</mi></mrow></msub><mo>=</mo><mi>X</mi></math></span> for all <span><math><mi>γ</mi><mo>∈</mo><mi>Γ</mi></math></span> and some Banach space <em>X</em>, we return to the well-explored ground of Köthe–Bochner sequence spaces <span><math><mi>E</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span>. Doing all this, we will reproduce, but sometimes also improve, essentially all existing results about the classical Kadets–Klee properties in Köthe–Bochner sequence spaces.</div></div>","PeriodicalId":55313,"journal":{"name":"Bulletin des Sciences Mathematiques","volume":"200 ","pages":"Article 103587"},"PeriodicalIF":1.3,"publicationDate":"2025-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143509435","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":"Adams-type inequalities with logarithmic weights in fractional dimensions and the existence of extremals","authors":"Rou Jiang ,&nbsp;Wenyan Xu ,&nbsp;Caifeng Zhang ,&nbsp;Maochun Zhu","doi":"10.1016/j.bulsci.2025.103586","DOIUrl":"10.1016/j.bulsci.2025.103586","url":null,"abstract":"<div><div>In this paper, we proved a sharp Adams-type inequality with logarithmic weights <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mi>β</mi></mrow></msub><mo>(</mo><mi>r</mi><mo>)</mo><mo>=</mo><msup><mrow><mo>(</mo><mi>log</mi><mo>⁡</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>r</mi></mrow></mfrac><mo>)</mo></mrow><mrow><mi>β</mi><mo>(</mo><mi>p</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></msup></math></span> or <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mi>β</mi></mrow></msub><mo>(</mo><mi>r</mi><mo>)</mo><mo>=</mo><msup><mrow><mo>(</mo><mi>log</mi><mo>⁡</mo><mfrac><mrow><mi>e</mi></mrow><mrow><mi>r</mi></mrow></mfrac><mo>)</mo></mrow><mrow><mi>β</mi><mo>(</mo><mi>p</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></msup></math></span>, <span><math><mi>β</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span> in the fractional dimensions. Furthermore, we show the existence of extremals for this kind of inequalities.</div></div>","PeriodicalId":55313,"journal":{"name":"Bulletin des Sciences Mathematiques","volume":"200 ","pages":"Article 103586"},"PeriodicalIF":1.3,"publicationDate":"2025-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143509434","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}