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On the Probabilistic Approximation in Reproducing Kernel Hilbert Spaces 论再现核希尔伯特空间中的概率逼近
arXiv - MATH - Functional Analysis Pub Date : 2024-09-18 DOI: arxiv-2409.11679
Dongwei Chen, Kai-Hsiang Wang
{"title":"On the Probabilistic Approximation in Reproducing Kernel Hilbert Spaces","authors":"Dongwei Chen, Kai-Hsiang Wang","doi":"arxiv-2409.11679","DOIUrl":"https://doi.org/arxiv-2409.11679","url":null,"abstract":"This paper generalizes the least square method to probabilistic approximation\u0000in reproducing kernel Hilbert spaces. We show the existence and uniqueness of\u0000the optimizer. Furthermore, we generalize the celebrated representer theorem in\u0000this setting, and especially when the probability measure is finitely\u0000supported, or the Hilbert space is finite-dimensional, we show that the\u0000approximation problem turns out to be a measure quantization problem. Some\u0000discussions and examples are also given when the space is infinite-dimensional\u0000and the measure is infinitely supported.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"66 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142249323","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
An optimization problem and point-evaluation in Paley-Wiener spaces 帕利-维纳空间中的优化问题和点评估
arXiv - MATH - Functional Analysis Pub Date : 2024-09-18 DOI: arxiv-2409.11963
Sarah May Instanes
{"title":"An optimization problem and point-evaluation in Paley-Wiener spaces","authors":"Sarah May Instanes","doi":"arxiv-2409.11963","DOIUrl":"https://doi.org/arxiv-2409.11963","url":null,"abstract":"We study the constant $mathscr{C}_p$ defined as the smallest constant $C$\u0000such that $|f(0)|^p leq C|f|_p^p$ holds for every function $f$ in the\u0000Paley-Wiener space $PW^p$. Brevig, Chirre, Ortega-Cerd`a, and Seip have\u0000recently shown that $mathscr{C}_p<p/2$ for all $p>2$. We improve this bound\u0000for $2<p leq 5$ by solving an optimization problem.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142249325","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On weighted Blaschke--Santalo and strong Brascamp--Lieb inequalities 论加权布拉什克--桑塔洛和强布拉什坎普--利布不等式
arXiv - MATH - Functional Analysis Pub Date : 2024-09-17 DOI: arxiv-2409.11503
Andrea Colesanti, Alexander Kolesnikov, Galyna Livshyts, Liran Rotem
{"title":"On weighted Blaschke--Santalo and strong Brascamp--Lieb inequalities","authors":"Andrea Colesanti, Alexander Kolesnikov, Galyna Livshyts, Liran Rotem","doi":"arxiv-2409.11503","DOIUrl":"https://doi.org/arxiv-2409.11503","url":null,"abstract":"In this paper, we study new extensions of the functional Blaschke-Santalo\u0000inequalities, and explore applications of such new inequalities beyond the\u0000classical setting of the standard Gaussian measure.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"44 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142249341","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Contractive Hilbert modules on quotient domains 商域上的收缩希尔伯特模块
arXiv - MATH - Functional Analysis Pub Date : 2024-09-17 DOI: arxiv-2409.11101
Shibananda Biswas, Gargi Ghosh, E. K. Narayanan, Subrata Shyam Roy
{"title":"Contractive Hilbert modules on quotient domains","authors":"Shibananda Biswas, Gargi Ghosh, E. K. Narayanan, Subrata Shyam Roy","doi":"arxiv-2409.11101","DOIUrl":"https://doi.org/arxiv-2409.11101","url":null,"abstract":"Let the complex reflection group $G(m,p,n)$ act on the unit polydisc $mathbb\u0000D^n$ in $mathbb C^n.$ A $boldsymbolTheta_n$-contraction is a commuting tuple\u0000of operators on a Hilbert space having\u0000$$overline{boldsymbolTheta}_n:={boldsymboltheta(z)=(theta_1(z),ldots,theta_n(z)):zinoverline{mathbb\u0000D}^n}$$ as a spectral set, where ${theta_i}_{i=1}^n$ is a homogeneous\u0000system of parameters associated to $G(m,p,n).$ A plethora of examples of\u0000$boldsymbolTheta_n$-contractions is exhibited. Under a mild hypothesis, it is\u0000shown that these $boldsymbolTheta_n$-contractions are mutually unitarily\u0000inequivalent. These inequivalence results are obtained concretely for the\u0000weighted Bergman modules under the action of the permutation groups and the\u0000dihedral groups. The division problem is shown to have negative answers for the\u0000Hardy module and the Bergman module on the bidisc. A Beurling-Lax-Halmos type\u0000representation for the invariant subspaces of $boldsymbolTheta_n$-isometries\u0000is obtained.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"191 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142249326","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On some multiple solutions for a $p(x)$-Laplace equation with supercritical growth 关于具有超临界增长的 $p(x)$ 拉普拉斯方程的一些多重解
arXiv - MATH - Functional Analysis Pub Date : 2024-09-17 DOI: arxiv-2409.10984
Lin Zhao
{"title":"On some multiple solutions for a $p(x)$-Laplace equation with supercritical growth","authors":"Lin Zhao","doi":"arxiv-2409.10984","DOIUrl":"https://doi.org/arxiv-2409.10984","url":null,"abstract":"We consider the multiplicity of solutions for the $p(x)$-Laplacian problems\u0000involving the supercritical Sobolev growth via Ricceri's principle. By means of\u0000the truncation combining with De Giorgi iteration, we can extend the result\u0000about subcritical and critical growth to the supercritical growth and obtain at\u0000least three solutions for the $p(x)$ Laplacian problem.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"105 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142268386","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Cesàro operators on the space of analytic functions with logarithmic growth 具有对数增长的解析函数空间上的塞萨罗算子
arXiv - MATH - Functional Analysis Pub Date : 2024-09-17 DOI: arxiv-2409.11371
José Bonet
{"title":"Cesàro operators on the space of analytic functions with logarithmic growth","authors":"José Bonet","doi":"arxiv-2409.11371","DOIUrl":"https://doi.org/arxiv-2409.11371","url":null,"abstract":"Continuity, compactness, the spectrum and ergodic properties of Ces`aro\u0000operators are investigated when they act on the space $VH(mathbb{D})$ of\u0000analytic functions with logarithmic growth on the open unit disc $mathbb{D}$\u0000of the complex plane. The space $VH(mathbb{D})$ is a countable inductive limit\u0000of weighted Banach spaces of analytic functions with compact linking maps. It\u0000was introduced and studied by Taskinen and also by Jasiczak.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"53 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142249324","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Section method and Frechet polynomials 截面法和弗雷谢特多项式
arXiv - MATH - Functional Analysis Pub Date : 2024-09-17 DOI: arxiv-2409.11204
Dan M Daianu
{"title":"Section method and Frechet polynomials","authors":"Dan M Daianu","doi":"arxiv-2409.11204","DOIUrl":"https://doi.org/arxiv-2409.11204","url":null,"abstract":"Using the section method we characterize the solutions $ f:Urightarrow Y$ of\u0000the following four equations begin{equation*} sumlimits_{i=0}^{n}left(\u0000-1right) ^{n-i}tbinom{n}{i}fleft( sqrt[m]{ u^{m}+iv^{m}}right) =left(\u0000n!right) fleft( vright) text{, } end{equation*} begin{equation*} fleft(\u0000uright) +sumlimits_{i=1}^{n+1}left( -1right) ^{i} tbinom{n+1}{i}fleft(\u0000sqrt[m]{u^{m}+iv^{m}}right) =0, end{equation*} begin{equation*}\u0000sumlimits_{i=0}^{n}left( -1right) ^{n-i}tbinom{n}{i}fleft( arcsin\u0000leftvert sin usin ^{i}vrightvert right) =left( n!right) fleft(\u0000vright) text{ and } end{equation*} begin{equation*} fleft( uright)\u0000+sumlimits_{i=1}^{n+1}left( -1right) ^{i}tbinom{n+1}{i% }fleft( arcsin\u0000leftvert sin usin ^{i}vrightvert right) =0, end{equation*} where $mgeq 2$ and $n$ are positive integers,$ Usubseteq mathbb{R} $ is a maximally relevant real domain and $left( Y,+right) $ is an $left(\u0000n!right) $ -divisible Abelian group.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142249328","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Operator orbit frames and frame-like Fourier expansions 算子轨道框架和类框架傅里叶展开
arXiv - MATH - Functional Analysis Pub Date : 2024-09-16 DOI: arxiv-2409.10706
Chad Berner, Eric. S. Weber
{"title":"Operator orbit frames and frame-like Fourier expansions","authors":"Chad Berner, Eric. S. Weber","doi":"arxiv-2409.10706","DOIUrl":"https://doi.org/arxiv-2409.10706","url":null,"abstract":"Frames in a Hilbert space that are generated by operator orbits are vastly\u0000studied because of the applications in dynamic sampling and signal recovery. We\u0000demonstrate in this paper a representation theory for frames generated by\u0000operator orbits that provides explicit constructions of the frame and the\u0000operator. It is known that the Kaczmarz algorithm for stationary sequences in\u0000Hilbert spaces generates a frame that arises from an operator orbit. In this\u0000paper, we show that every frame generated by operator orbits in any Hilbert\u0000space arises from the Kaczmarz algorithm. Furthermore, we show that the\u0000operators generating these frames are similar to rank one perturbations of\u0000unitary operators. After this, we describe a large class of operator orbit\u0000frames that arise from Fourier expansions for singular measures. Moreover, we\u0000classify all measures that possess frame-like Fourier expansions arising from\u0000two-sided operator orbit frames. Finally, we show that measures that possess\u0000frame-like Fourier expansions arising from two-sided operator orbits are\u0000weighted Lebesgue measure with weight satisfying a weak $A_{2}$ condition, even\u0000in the non-frame case. We also use these results to classify measures with\u0000other types of frame-like Fourier expansions.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"38 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142249327","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Shift-cyclicity in analytic function spaces 解析函数空间的移环性
arXiv - MATH - Functional Analysis Pub Date : 2024-09-16 DOI: arxiv-2409.10224
Jeet Sampat
{"title":"Shift-cyclicity in analytic function spaces","authors":"Jeet Sampat","doi":"arxiv-2409.10224","DOIUrl":"https://doi.org/arxiv-2409.10224","url":null,"abstract":"In this survey, we consider Banach spaces of analytic functions in one and\u0000several complex variables for which: (i) polynomials are dense, (ii)\u0000point-evaluations on the domain are bounded linear functionals, and (iii) the\u0000shift operators are bounded for each variable. We discuss the problem of\u0000determining the shift-cyclic functions in such a space, i.e., functions whose\u0000polynomial multiples form a dense subspace. The problem of determining\u0000shift-cyclic functions in certain analytic function spaces is known to be\u0000intimately connected to some deep problems in other areas of mathematics, such\u0000as the dilation completeness problem and even the Riemann hypothesis. What makes determining shift-cyclic functions so difficult is that often we\u0000need to employ techniques that are specific to the space in consideration. We\u0000therefore cover several different function spaces that have frequently appeared\u0000in the past such as the Hardy spaces, Dirichlet-type spaces, complete Pick\u0000spaces and Bergman spaces. We highlight the similarities and the differences\u0000between shift-cyclic functions among these spaces and list some important\u0000general properties that shift-cyclic functions in any given analytic function\u0000space must share. Throughout this discussion, we also motivate and provide a\u0000large list of open problems related to shift-cyclicity.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142249330","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Few operators on Banach spaces $C_0(Ltimes L)$ 巴拿赫空间上的少数算子 $C_0(Ltimes L)$
arXiv - MATH - Functional Analysis Pub Date : 2024-09-16 DOI: arxiv-2409.10477
Leandro Candido
{"title":"Few operators on Banach spaces $C_0(Ltimes L)$","authors":"Leandro Candido","doi":"arxiv-2409.10477","DOIUrl":"https://doi.org/arxiv-2409.10477","url":null,"abstract":"Using Ostaszewski's $clubsuit$-principle, we construct a non-metrizable,\u0000locally compact, scattered space $L$ in which the operators on the Banach space\u0000$C_0(L times L)$ exhibit a remarkably simple structure. We provide a detailed\u0000analysis and, through a series of decomposition steps, offer an explicit\u0000characterization of all operators on $C_0(L times L)$.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142249331","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","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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