This paper generalizes the least square method to probabilistic approximation in reproducing kernel Hilbert spaces. We show the existence and uniqueness of the optimizer. Furthermore, we generalize the celebrated representer theorem in this setting, and especially when the probability measure is finitely supported, or the Hilbert space is finite-dimensional, we show that the approximation problem turns out to be a measure quantization problem. Some discussions and examples are also given when the space is infinite-dimensional and the measure is infinitely supported.
{"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}
We study the constant $mathscr{C}_p$ defined as the smallest constant $C$ such that $|f(0)|^p leq C|f|_p^p$ holds for every function $f$ in the Paley-Wiener space $PW^p$. Brevig, Chirre, Ortega-Cerd`a, and Seip have recently shown that $mathscr{C}_p
{"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}
Andrea Colesanti, Alexander Kolesnikov, Galyna Livshyts, Liran Rotem
In this paper, we study new extensions of the functional Blaschke-Santalo inequalities, and explore applications of such new inequalities beyond the classical setting of the standard Gaussian measure.
{"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}
Shibananda Biswas, Gargi Ghosh, E. K. Narayanan, Subrata Shyam Roy
Let the complex reflection group $G(m,p,n)$ act on the unit polydisc $mathbb D^n$ in $mathbb C^n.$ A $boldsymbolTheta_n$-contraction is a commuting tuple of operators on a Hilbert space having $$overline{boldsymbolTheta}_n:={boldsymboltheta(z)=(theta_1(z),ldots,theta_n(z)):zinoverline{mathbb D}^n}$$ as a spectral set, where ${theta_i}_{i=1}^n$ is a homogeneous system of parameters associated to $G(m,p,n).$ A plethora of examples of $boldsymbolTheta_n$-contractions is exhibited. Under a mild hypothesis, it is shown that these $boldsymbolTheta_n$-contractions are mutually unitarily inequivalent. These inequivalence results are obtained concretely for the weighted Bergman modules under the action of the permutation groups and the dihedral groups. The division problem is shown to have negative answers for the Hardy module and the Bergman module on the bidisc. A Beurling-Lax-Halmos type representation for the invariant subspaces of $boldsymbolTheta_n$-isometries is obtained.
{"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}
We consider the multiplicity of solutions for the $p(x)$-Laplacian problems involving the supercritical Sobolev growth via Ricceri's principle. By means of the truncation combining with De Giorgi iteration, we can extend the result about subcritical and critical growth to the supercritical growth and obtain at least three solutions for the $p(x)$ Laplacian problem.
我们通过里切利原理考虑了涉及超临界索波列夫增长的 $p(x)$ 拉普拉奇问题解的多重性。通过截断与 De Giorgi 迭代相结合的方法，我们可以将亚临界和临界增长的结果扩展到超临界增长，并得到 $p(x)$ 拉普拉斯问题的至少三个解。
{"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}
Continuity, compactness, the spectrum and ergodic properties of Ces`aro operators are investigated when they act on the space $VH(mathbb{D})$ of analytic functions with logarithmic growth on the open unit disc $mathbb{D}$ of the complex plane. The space $VH(mathbb{D})$ is a countable inductive limit of weighted Banach spaces of analytic functions with compact linking maps. It was introduced and studied by Taskinen and also by Jasiczak.
{"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}
Frames in a Hilbert space that are generated by operator orbits are vastly studied because of the applications in dynamic sampling and signal recovery. We demonstrate in this paper a representation theory for frames generated by operator orbits that provides explicit constructions of the frame and the operator. It is known that the Kaczmarz algorithm for stationary sequences in Hilbert spaces generates a frame that arises from an operator orbit. In this paper, we show that every frame generated by operator orbits in any Hilbert space arises from the Kaczmarz algorithm. Furthermore, we show that the operators generating these frames are similar to rank one perturbations of unitary operators. After this, we describe a large class of operator orbit frames that arise from Fourier expansions for singular measures. Moreover, we classify all measures that possess frame-like Fourier expansions arising from two-sided operator orbit frames. Finally, we show that measures that possess frame-like Fourier expansions arising from two-sided operator orbits are weighted Lebesgue measure with weight satisfying a weak $A_{2}$ condition, even in the non-frame case. We also use these results to classify measures with other types of frame-like Fourier expansions.
{"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}
In this survey, we consider Banach spaces of analytic functions in one and several complex variables for which: (i) polynomials are dense, (ii) point-evaluations on the domain are bounded linear functionals, and (iii) the shift operators are bounded for each variable. We discuss the problem of determining the shift-cyclic functions in such a space, i.e., functions whose polynomial multiples form a dense subspace. The problem of determining shift-cyclic functions in certain analytic function spaces is known to be intimately connected to some deep problems in other areas of mathematics, such as the dilation completeness problem and even the Riemann hypothesis. What makes determining shift-cyclic functions so difficult is that often we need to employ techniques that are specific to the space in consideration. We therefore cover several different function spaces that have frequently appeared in the past such as the Hardy spaces, Dirichlet-type spaces, complete Pick spaces and Bergman spaces. We highlight the similarities and the differences between shift-cyclic functions among these spaces and list some important general properties that shift-cyclic functions in any given analytic function space must share. Throughout this discussion, we also motivate and provide a large list of open problems related to shift-cyclicity.
{"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}
Using Ostaszewski's $clubsuit$-principle, we construct a non-metrizable, locally compact, scattered space $L$ in which the operators on the Banach space $C_0(L times L)$ exhibit a remarkably simple structure. We provide a detailed analysis and, through a series of decomposition steps, offer an explicit characterization of all operators on $C_0(L times L)$.
利用奥斯塔谢夫斯基的$clubsuit$原理，我们构造了一个非三元的、局部紧凑的、分散的空间$L$，其中巴拿赫空间$C_0(L times L)$上的算子表现出非常简单的结构。我们提供了详细的分析，并通过一系列分解步骤，提供了$C_0(L times L)$上所有算子的解释性特征。
{"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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：