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Approximating Reproducing Kernel Hilbert Space Functions by Bernstein Operators 用伯恩斯坦算子逼近重现核希尔伯特空间函数
IF 2.2 3区 数学
Results in Mathematics Pub Date : 2024-08-30 DOI: 10.1007/s00025-024-02253-w
Han Feng, Sonia Y. W. Hui, Ruohan Shen
{"title":"Approximating Reproducing Kernel Hilbert Space Functions by Bernstein Operators","authors":"Han Feng, Sonia Y. W. Hui, Ruohan Shen","doi":"10.1007/s00025-024-02253-w","DOIUrl":"https://doi.org/10.1007/s00025-024-02253-w","url":null,"abstract":"<p>Motivated by kernel methods in machine learning theory, we study the uniform approximation of functions from reproducing kernel Hilbert spaces by Bernstein operators. Rates of approximation are provided in terms of the function norm in the reproducing kernel Hilbert space. A case study of contracting properties of the Bernstein operators is also presented.</p>","PeriodicalId":54490,"journal":{"name":"Results in Mathematics","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142194344","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}
引用次数: 0
Uniform Convergence of Global Least Energy Solutions to Dirichlet Systems in Non-reflexive Orlicz–Sobolev Spaces 非反折 Orlicz-Sobolev 空间中 Dirichlet 系统的全局最小能量解的均匀收敛性
IF 2.2 3区 数学
Results in Mathematics Pub Date : 2024-08-30 DOI: 10.1007/s00025-024-02270-9
Grey Ercole, Giovany M. Figueiredo, Abdolrahman Razani
{"title":"Uniform Convergence of Global Least Energy Solutions to Dirichlet Systems in Non-reflexive Orlicz–Sobolev Spaces","authors":"Grey Ercole, Giovany M. Figueiredo, Abdolrahman Razani","doi":"10.1007/s00025-024-02270-9","DOIUrl":"https://doi.org/10.1007/s00025-024-02270-9","url":null,"abstract":"<p>We prove that for each <span>(pin (1,infty ))</span> the energy functional associated with the Dirichlet system </p><span>$$begin{aligned} left{ begin{array}{lll} -{text {div}}(phi _{p}(left| nabla uright| )nabla u)=partial _{1}F(u,v) &amp; textrm{in} &amp; Omega , -{text {div}}(phi _{p}(left| nabla vright| )nabla v)=partial _{2}F(u,v) &amp; textrm{in} &amp; Omega , u=v=0 &amp; textrm{on} &amp; partial Omega , end{array} right. end{aligned}$$</span><p>admits at least one global, nonnegative minimizer <span>((u_{p},v_{p})in W_{0}^{Phi _{p}}(Omega )times W_{0}^{Phi _{p}}(Omega ))</span> which converges uniformly on <span>(overline{Omega })</span> to <span>((d_{Omega },d_{Omega }),)</span> as <span>(prightarrow infty )</span>. Here <span>(Phi _{p}(t):=int _{0}^{t}sphi _{p}(left| sright| )textrm{d}s)</span> and <span>(d_{Omega })</span> stands for the distance function to the boundary <span>(partial Omega )</span>.\u0000</p>","PeriodicalId":54490,"journal":{"name":"Results in Mathematics","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142194347","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}
引用次数: 0
Josephy’s Theorem, Revisited 重温约瑟夫定理
IF 2.2 3区 数学
Results in Mathematics Pub Date : 2024-08-30 DOI: 10.1007/s00025-024-02265-6
Daria Bugajewska, Piotr Kasprzak
{"title":"Josephy’s Theorem, Revisited","authors":"Daria Bugajewska, Piotr Kasprzak","doi":"10.1007/s00025-024-02265-6","DOIUrl":"https://doi.org/10.1007/s00025-024-02265-6","url":null,"abstract":"<p>The main goal of this note is to characterize the necessary and sufficient conditions for a composition operator to act between spaces of mappings of bounded Wiener variation in a normed-valued setting. The necessary and sufficient conditions for local boundedness of such operators are also discussed.</p>","PeriodicalId":54490,"journal":{"name":"Results in Mathematics","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142194345","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}
引用次数: 0
Secant Distributions of Unitals 单位的正割分布
IF 2.2 3区 数学
Results in Mathematics Pub Date : 2024-08-26 DOI: 10.1007/s00025-024-02261-w
Mustafa Gezek
{"title":"Secant Distributions of Unitals","authors":"Mustafa Gezek","doi":"10.1007/s00025-024-02261-w","DOIUrl":"https://doi.org/10.1007/s00025-024-02261-w","url":null,"abstract":"<p>Let <i>U</i> be a unital embedded in a projective plane <span>(Pi )</span> of order <span>(q^2)</span>. For <span>(Rin U)</span>, let <span>(s_R)</span> and <span>(t_R)</span> be a secant line through <i>R</i> and the tangent line to <i>U</i> at point <i>R</i>, respectively. If the tangent lines to <i>U</i>, passing through the points in <span>(s_Rcap U)</span>, intersect at a single point on <span>(t_R)</span>, then <span>(s_R)</span> is referred to as a secant line satisfying the desired property. If <span>(n_i)</span> of the points of <i>U</i> have exactly <span>(m_i)</span> secant lines satisfying the desired property, then </p><span>$$begin{aligned} m_1^{n_1}, m_2^{n_2}, cdots end{aligned}$$</span><p>is called the secant distribution of <i>U</i>, where <span>(sum n_i=q^3+1)</span>, and <span>(0le m_ile q^2)</span>. In this article, we show that collinear pedal sets of a unital <i>U</i> plays an important role in the secant distribution of <i>U</i>. Formulas for secant distributions of unitals having <span>(0,1,q^2,)</span> or <span>(q^2+q)</span> special points are provided. Statistics regarding to secant distributions of unitals embedded in planes of orders <span>(q^2le 25)</span> are presented. Some open problems related to secant distributions of unitals having specific number of collinear pedal sets are discussed. </p>","PeriodicalId":54490,"journal":{"name":"Results in Mathematics","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142194348","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}
引用次数: 0
Positive Solutions for Convective Double Phase Problems 对流双相问题的正解
IF 2.2 3区 数学
Results in Mathematics Pub Date : 2024-08-26 DOI: 10.1007/s00025-024-02262-9
Nikolao S. Papageorgiou, Zijia Peng
{"title":"Positive Solutions for Convective Double Phase Problems","authors":"Nikolao S. Papageorgiou, Zijia Peng","doi":"10.1007/s00025-024-02262-9","DOIUrl":"https://doi.org/10.1007/s00025-024-02262-9","url":null,"abstract":"<p>We consider a nonlinear Dirichlet problem driven by the double phase differential operator and a reaction that has the competing effects of a parametric concave term and of a convective perturbation. Using truncation and comparison techniques and the theory of nonlinear operators of monotone type, we show that for all small values of the parameter, the problem has a bounded positive solution.</p>","PeriodicalId":54490,"journal":{"name":"Results in Mathematics","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142194375","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}
引用次数: 0
Restriction Theorems for the p-Analog of the Fourier–Stieltjes Algebra 傅里叶-斯蒂尔杰斯代数 p-Analog 的限制定理
IF 2.2 3区 数学
Results in Mathematics Pub Date : 2024-08-26 DOI: 10.1007/s00025-024-02263-8
Arvish Dabra, N. Shravan Kumar
{"title":"Restriction Theorems for the p-Analog of the Fourier–Stieltjes Algebra","authors":"Arvish Dabra, N. Shravan Kumar","doi":"10.1007/s00025-024-02263-8","DOIUrl":"https://doi.org/10.1007/s00025-024-02263-8","url":null,"abstract":"<p>For a locally compact group <i>G</i> and <span>(1&lt; p &lt; infty ,)</span> let <span>(B_{p}(G))</span> denote the <i>p</i>-analog of the Fourier–Stieltjes algebra <span>(B(G) , (text {or} , B_2(G)))</span>. Let <span>(r: B_{p}(G) rightarrow B_p(H))</span> be the restriction map given by <span>(r(u) = u|_H)</span> for any closed subgroup <i>H</i> of <i>G</i>. In this article, we prove that the restriction map <i>r</i> is a surjective isometry for any open subgroup <i>H</i> of <i>G</i>. Further, we show that the range of the map <i>r</i> is dense in <span>(B_p(H))</span> when <i>H</i> is either a compact normal subgroup of <i>G</i> or compact subgroup of an [SIN]<span>(_H)</span>-group.</p>","PeriodicalId":54490,"journal":{"name":"Results in Mathematics","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142194376","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}
引用次数: 0
Analytic Kramer Sampling and Quasi Lagrange-Type Interpolation in Vector Valued RKHS 矢量值 RKHS 中的解析克拉默采样和准拉格朗日型插值法
IF 2.2 3区 数学
Results in Mathematics Pub Date : 2024-08-23 DOI: 10.1007/s00025-024-02260-x
Subhankar Mahapatra, Santanu Sarkar
{"title":"Analytic Kramer Sampling and Quasi Lagrange-Type Interpolation in Vector Valued RKHS","authors":"Subhankar Mahapatra, Santanu Sarkar","doi":"10.1007/s00025-024-02260-x","DOIUrl":"https://doi.org/10.1007/s00025-024-02260-x","url":null,"abstract":"<p>This paper discusses an abstract Kramer sampling theorem for functions within a reproducing kernel Hilbert space (RKHS) of vector valued holomorphic functions. Additionally, we extend the concept of quasi Lagrange-type interpolation for functions within a RKHS of vector valued entire functions. The dependence of having quasi Lagrange-type interpolation on an invariance condition under the generalized backward shift operator has also been discussed. Furthermore, the paper establishes the connection between quasi Lagrange-type interpolation, operator of multiplication by the independent variable, and de Branges spaces of vector valued entire functions.</p>","PeriodicalId":54490,"journal":{"name":"Results in Mathematics","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142194377","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}
引用次数: 0
An HK $$_r$$ -Integrable Function Which is P $$_s$$ -Integrable for no s 对于无 s 的 P $$_s$$ 可包容的 HK $$_r$$ 可包容函数
IF 2.2 3区 数学
Results in Mathematics Pub Date : 2024-08-19 DOI: 10.1007/s00025-024-02251-y
Piotr Sworowski
{"title":"An HK $$_r$$ -Integrable Function Which is P $$_s$$ -Integrable for no s","authors":"Piotr Sworowski","doi":"10.1007/s00025-024-02251-y","DOIUrl":"https://doi.org/10.1007/s00025-024-02251-y","url":null,"abstract":"<p>Given arbitrary <span>(rge 1)</span>, we construct an HK<span>(_r)</span>-integrable function which is not P<span>(_1)</span>-integrable. This is an extension of a recently published construction [Musial, P., Skvortsov, V., Tulone, F.: The HK<span>(_r)</span>-integral is not contained in the P<span>(_r)</span>-integral. Proceedings of the American Mathematical Society <b>150</b>(5), 2107–2114 (2022)].</p>","PeriodicalId":54490,"journal":{"name":"Results in Mathematics","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142194378","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}
引用次数: 0
New Asymptotic Evaluations for Sequences of Linear Positive Operators on $$C_{2pi }left( {{mathbb {R}}}right) $$ 线性正算子序列在 $$C_{2pi }left( {{mathbb {R}}}right) $$ 上的新渐近评估
IF 2.2 3区 数学
Results in Mathematics Pub Date : 2024-08-17 DOI: 10.1007/s00025-024-02257-6
Dumitru Popa
{"title":"New Asymptotic Evaluations for Sequences of Linear Positive Operators on $$C_{2pi }left( {{mathbb {R}}}right) $$","authors":"Dumitru Popa","doi":"10.1007/s00025-024-02257-6","DOIUrl":"https://doi.org/10.1007/s00025-024-02257-6","url":null,"abstract":"<p>In the paper we give new asymptotic evaluations for sequences of linear positive operators <span>(V_{n}:C_{2pi }left( {{mathbb {R}}}right) rightarrow C_{2pi }left( {{mathbb {R}}}right) )</span>. Our method of the proof is entirely different than those known in this area. As applications we complete and extend known asymptotic evaluations in this topic.</p>","PeriodicalId":54490,"journal":{"name":"Results in Mathematics","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142194380","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}
引用次数: 0
Non-coercive Neumann Boundary Control Problems 非强制 Neumann 边界控制问题
IF 2.2 3区 数学
Results in Mathematics Pub Date : 2024-08-17 DOI: 10.1007/s00025-024-02255-8
Thomas Apel, Mariano Mateos, Arnd Rösch
{"title":"Non-coercive Neumann Boundary Control Problems","authors":"Thomas Apel, Mariano Mateos, Arnd Rösch","doi":"10.1007/s00025-024-02255-8","DOIUrl":"https://doi.org/10.1007/s00025-024-02255-8","url":null,"abstract":"<p>The article examines a linear-quadratic Neumann control problem that is governed by a non-coercive elliptic equation. Due to the non-self-adjoint nature of the linear control-to-state operator, it is necessary to independently study both the state and adjoint state equations. The article establishes the existence and uniqueness of solutions for both equations, with minimal assumptions made about the problem’s data. Next, the regularity of these solutions is studied in three frameworks: Hilbert–Sobolev spaces, Sobolev–Slobodeckiĭ spaces, and weighted Sobolev spaces. These regularity results enable a numerical analysis of the finite element approximation of both the state and adjoint state equations. The results cover both convex and non-convex domains and quasi-uniform and graded meshes. Finally, the optimal control problem is analyzed and discretized. Existence and uniqueness of the solution, first-order optimality conditions, and error estimates for the finite element approximation of the control are obtained. Numerical experiments confirming these results are included.</p>","PeriodicalId":54490,"journal":{"name":"Results in Mathematics","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142194379","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}
引用次数: 0
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