Alan Rufty
A reproducing kernel Hilbert space (RKHS) has four well-known easily derived properties. Since these properties are usually not emphasized as a simple means of gaining insight into RKHS structure, they are singled out and proved in this brief article.
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This paper is devoted to the study of vector valued reproducing kernel Hilbert spaces. We focus on reproducing kernels in vector-valued reproducing kernel Hilbert spaces. In particular we extend reproducing kernels to relative reproducing kernels and prove some theorems in this subject.
This paper is devoted to the study of reproducing kernel Hilbert spaces. We focus on multipliers of reproducing kernel Banach and Hilbert spaces. In particular we tried to extend this concept and prove some theorems.
In this paper, we give some results concerning atomic decompositions for operators on reproducing kernel Hilbert spaces, using frame theory techniques. We provide also generalizations for atomic decompositions of some theorems for reproducing kernel Hilbert spaces. In particular, we obtain atomic decomposition results for operators on Bergman spaces and Fock spaces in a simple manner.
In this note, we describe the backward shift invariant subspaces for a large class of reproducing kernel Hilbert spaces. This class includes in particular de Branges-Rovnyak spaces (the non-extreme case) and the range space of co-analytic Toeplitz operators.
We consider conditions on a given system $\mathcal{F}$ of vectors in Hilbert space $\mathcal{H}$, forming a frame, which turn $\mathcal{H}$ into a reproducing kernel Hilbert space. It is assumed that the vectors in $\mathcal{F}$ are functions on some set $Ω$. We then identify conditions on these functions which automatically give $\mathcal{H}$ the structure of a reproducing kernel Hilbert space of functions on $Ω$. We further give an explicit formula for the kernel, and for the corresponding isometric isomorphism. Applications are given to Hilbert spaces associated to families of Gaussian proc…
In this paper, we introduce the notion of reproducing kernel Hilbert spaces for graphs and the Gram matrices associated with them. Our aim is to investigate the Gram matrices of reproducing kernel Hilbert spaces. We provide several bounds on the entries of the Gram matrices of reproducing kernel Hilbert spaces and characterize the graphs which attain our bounds.
This paper is devoted to the study of vector valued reproducing kernel Hilbert spaces. We focus on two aspects: vector valued feature maps and universal kernels. In particular we characterize the structure of translation invariant kernels on abelian groups and we relate it to the universality problem.
A well-known theorem due to R. E. Curto and N. Salinas gives a necessary and sufficient condition for the unitary equivalence of commuting tuples of bounded linear operators acting on reproducing kernel Hilbert spaces. Inspired by this theorem, we obtain a different but equivalent criterion for the unitary equivalence of operators acting on reproducing kernel Hilbert spaces. As an application, we describe the structure of intertwining operator and prove that the decomposition of Cowen-Douglas operators is unique up to unitary equivalence.
The aim of this paper is to present a unified framework in the setting of Hilbert $C^*$-modules for the scalar- and vector-valued reproducing kernel Hilbert spaces and $C^*$-valued reproducing kernel spaces. We investigate conditionally negative definite kernels with values in the $C^*$-algebra of adjointable operators acting on a Hilbert $C^*$-module. In addition, we show that there exists a two-sided connection between positive definite kernels and reproducing kernel Hilbert $C^*$-modules. Furthermore, we explore some conditions under which a function is in the reproducing kernel module and …
We give two new global and algorithmic constructions of the reproducing kernel Hilbert space associated to a positive definite kernel. We further present ageneral positive definite kernel setting using bilinear forms, and we provide new examples. Our results cover the case of measurable positive definite kernels, and we give applications to both stochastic analysisand metric geometry and provide a number of examples.
Let $(X,μ)$ be a strictly-positive Borel measure space. We show that the modes of convergence in a reproducing kernel Hilbert (RKHS) space, pointwise, weak and strong are all equivalents. From this we describe some important consequences such as an association with positive operators and positive definite kernels and a compact embedding condition for a RKHS in $L^{2}(X,μ)$.
Several upper and lower bounds of the Davis-Wielandt-Berezin radius of bounded linear operators defined on a reproducing kernel Hilbert space are given. Further, an inequality involving the Berezin number and the Davis-Wielandt-Berezin radius for the sum of two bounded linear operators is obtained, namely, if $A $ and $B$ are reproducing kernel Hilbert space operators, then $$η(A+B) \leq η(A)+η(B)+\textbf{ber}(A^*B+B^*A),$$ where $η(\cdot)$ and $\textbf{ber}(\cdot)$ are the Davis-Wielandt-Berezin radius and the Berezin number, respectively.
We prove two new equivalences of the Feichtinger conjecture that involve reproducing kernel Hilbert spaces. We prove that if for every Hilbert space, contractively contained in the Hardy space, each Bessel sequence of normalized kernel functions can be partitioned into finitely many Riesz basic sequences, then a general bounded Bessel sequence in an arbitrary Hilbert space can be partitioned into finitely many Riesz basic sequences. In addition, we examine some of these spaces and prove that for these spaces bounded Bessel sequences of normalized kernel functions are finite unions of Riesz bas…
To help understand various reproducing kernels used in applied sciences, we investigate the inclusion relation of two reproducing kernel Hilbert spaces. Characterizations in terms of feature maps of the corresponding reproducing kernels are established. A full table of inclusion relations among widely-used translation invariant kernels is given. Concrete examples for Hilbert-Schmidt kernels are presented as well. We also discuss the preservation of such a relation under various operations of reproducing kernels. Finally, we briefly discuss the special inclusion with a norm equivalence.
We demonstrate that a reproducing kernel Hilbert or Banach space of functions on a separable absolute Borel space or an analytic subset of a Polish space is separable if it possesses a Borel measurable feature map.
The geometry of spaces with indefinite inner product, known also as Krein spaces, is a basic tool for developing Operator Theory therein. In the present paper we establish a link between this geometry and the algebraic theory of *-semigroups. It goes via the positive definite functions and related to them reproducing kernel Hilbert spaces. Our concern is in describing properties of elements of the semigroup which determine shift operators which serve as Pontryagin fundamental symmetries
We characterize the reproducing kernel Hilbert spaces whose elements are $p$-integrable functions in terms of the boundedness of the integral operator whose kernel is the reproducing kernel. Moreover, for $p=2$ we show that the spectral decomposition of this integral operator gives a complete description of the reproducing kernel.
Given an uncountable, compact metric space, we show that there exists no reproducing kernel Hilbert space that contains the space of all continuous functions on this compact space.
This paper considers different facets of the interplay between reproducing kernel Hilbert spaces (RKHS) and stable analysis/synthesis processes: First, we analyze the structure of the reproducing kernel of a RKHS using frames and reproducing pairs. Second, we present a new approach to prove the result that finite redundancy of a continuous frame implies atomic structure of the underlying measure space. Our proof uses the RKHS structure of the range of the analysis operator. This in turn implies that all the attempts to extend the notion of Riesz basis to general measure spaces are fruitless si…
On the Probabilistic Approximation in Reproducing Kernel Hilbert Spaces
本文研究了再生核 Hilbert 空间中的概率函数逼近问题，证明了优化器的存在唯一性，推广了表示定理，并揭示了其在有限支撑与无限支撑下与测度量化和采样理论的联系。
We study a reproducing kernel Hilbert space of functions defined on the positive integers and associated to the binomial coefficients. We introduce two transforms, which allow us to develop a related harmonic analysis in this Hilbert space. Finally, we mention connections with the theory of discrete analytic functions and with the quantum case.
We give a simple proof of the so called reproducing kernel thesis for Hankel operators
Construction of Reproducing Kernel Hilbert Spaces Associated with the Laplace--Bessel Operator
该研究受贝塞尔算子偏微分方程问题启发，引入新的再生核希尔伯特空间，此空间是分数加权索伯列夫空间且有正定核，还获得奇异泊松方程的简单实用近似解。
In this paper we study two separate problems on interpolation. We first give some new equivalences of Stout's Theorem on necessary and sufficient conditions for a sequence of points to be an interpolating sequence on a finite open Riemann surface. We next turn our attention to the question of interpolation for reproducing kernel Hilbert spaces on the polydisc and provide a collection of equivalent statements about when it is possible to interpolation in the Schur-Agler class of the associated reproducing kernel Hilbert space.
We present necessary and sufficient conditions to hold true a Kramer type sampling theorem over semi-inner product reproducing kernel Banach spaces. Under some sampling-type hypotheses over a sequence of functions on these Banach spaces it results necessary that such sequence must be a $X_d$-Riesz basis and a sampling basis for the space. These results are a generalization of some already known sampling theorems over reproducing kernel Hilbert spaces.
Using the smallest eigenvalues of Hankel forms associated with a multidimensional moment problem, we establish a condition equivalent to the existence of a reproducing kernel. This result is a multivariate analogue of Berg, Chen,and Ismail's 2002 result. We also present a class of measures for which the existence of a reproducing kernel implies indeterminacy.
This paper studies the construction of a refinement kernel for a given operator-valued reproducing kernel such that the vector-valued reproducing kernel Hilbert space of the refinement kernel contains that of the given one as a subspace. The study is motivated from the need of updating the current operator-valued reproducing kernel in multi-task learning when underfitting or overfitting occurs. Numerical simulations confirm that the established refinement kernel method is able to meet this need. Various characterizations are provided based on feature maps and vector-valued integral representat…
We present sampling theorems for reproducing kernel Banach spaces on Lie groups. Recent approaches to this problem rely on integrability of the kernel and its local oscillations. In this paper we replace the integrability conditions by requirements on the derivatives of the reproducing kernel. The results are then used to obtain frames and atomic decompositions for Banach spaces of distributions stemming from a cyclic representation, and it is shown that this is particularly easy, when the cyclic vector is a Gårding vector for a square integrable representation.
Characterization of the reproducing structure of the Bessel potential spaces beyond $p=2$
本文研究了Bessel势空间在p≠2时的再生结构特征，通过分析Matèrn核作为再生核的性质，确定了不同参数下Bessel势空间的再生核条件。
We generalize the orthonormal basis for the Gaussian RKHS described in \cite{MinhGaussian2010} to an infinite, continuously parametrized, family of orthonormal bases, along with some implications. The proofs are direct generalizations of those in \cite{MinhGaussian2010}.
Properties of reproducing kernel Hilbert spaces of a group action
本文在紧豪斯多夫空间$X$上引入等价关系并建立其三个等价定义，探讨了$X$的等价类与群的关系稳定子群的对应关系，研究了群作用的再生核希尔伯特空间的性质。
Applications of Reproducing Kernels in composition operators
本文通过再现核方法研究复合算子，揭示其在加权 Hardy 空间中的结构特性及分类结果，并简化了若干已知结论的证明过程。
An algebra structure for reproducing kernel Hilbert spaces
本文在再生核希尔伯特空间上定义代数结构，等价于权的次卷积性，证明该类在张量积和拉回下封闭，且谱函子像包含所有紧子空间。
We study reproducing kernels, and associated reproducing kernel Hilbert spaces (RKHSs) $\mathscr{H}$ over infinite, discrete and countable sets $V$. In this setting we analyze in detail the distributions of the corresponding Dirac point-masses of $V$. Illustrations include certain models from neural networks: An Extreme Learning Machine (ELM) is a neural network-configuration in which a hidden layer of weights are randomly sampled, and where the object is then to compute resulting output. For RKHSs $\mathscr{H}$ of functions defined on a prescribed countable infinite discrete set $V$, we chara…
We study kernel functions, and associated reproducing kernel Hilbert spaces $\mathscr{H}$ over infinite, discrete and countable sets $V$. Numerical analysis builds discrete models (e.g., finite element) for the purpose of finding approximate solutions to boundary value problems; using multiresolution-subdivision schemes in continuous domains. In this paper, we turn the tables: our object of study is realistic infinite discrete models in their own right; and we then use an analysis of suitable continuous counterpart problems, but now serving as a tool for obtaining solutions in the discrete wor…
We characterize those generating functions k that produce weighted Hardy spaces of the unit disk D supporting nontrivial Hermitian weighted composition operators. Our characterization shows that the spaces associated with the "classical reproducing kernels," as well as certain natural extensions of these spaces, are precisely those that are hospitable to Hermitian weighted composition operators. It also leads to a refinement of a necessary condition for a weighted composition to be Hermitian, obtained recently by Cowen, Gunatillake, and Ko, into one that is both necessary and sufficient.
Reproducing Kernel Hilbert Spaces and entropy Kolmogorov numbers on compact Lie Groups
本文研究了紧致李群上再生核希尔伯特空间的熵柯尔莫戈罗夫数的渐近估计，探讨了其与连续函数空间嵌入关系中的关键数学性质。
Commutators of the Unilateral Shift and Adjoint for Reproducing Kernel Hilbert Spaces on the Disk
该文将Berger-Shaw迹公式从Hardy空间推广至单位圆盘上更广的旋转不变再生核希尔伯特空间，并给出内积实例及加权移位扩展。
A Survey on Reproducing Kernel Krein Spaces
本文综述了再生核 Krein 空间及其与算子值 Hermitian 核的相互作用，利用线性化与 Kolmogorov 分解等抽象概念探讨存在性与唯一性，并强调其算子值域特征及 Szego 核的普遍性。
Complementing earlier results on dynamics of unilateral weighted shifts, we obtain a sufficient (but not necessary, with supporting examples) condition for hypercyclicity, mixing and chaos for $M_z^*$, the adjoint of $M_z$, on vector-valued analytic reproducing kernel Hilbert spaces $\mathcal{H}$ in terms of the derivatives of kernel functions on the open unit disc $\mathbb{D}$ in $\mathbb{C}$. Here $M_z$ denotes the multiplication operator by the coordinate function $z$, that is \[ (M_z f) (w) = w f(w), \] for all $f \in \mathcal{H}$ and $w \in \mathbb{D}$. We analyze the special case of quas…
H-harmonic reproducing kernels on the ball
本文研究了n维单位球上$H$-调和函数空间的Szegő再生核，推导了基于Exton三重超几何函数的显式级数展开，并证明其可表示为超几何函数的有限和及第一Appell超几何函数形式，同时给出了加权Bergman核的级数展开。
We study reproducing kernel Hilbert spaces (RKHS) on a Riemannian manifold. In particular, we discuss under which condition Sobolev spaces are RKHS and characterize their reproducing kernels. Further, we introduce and discuss a class of smoother RKHS that we call diffusion spaces. We illustrate the general results with a number of detailed examples.
Improved upper bounds for the Berezin numbers of operators on reproducing kernel Hilbert spaces
研究再生核希尔伯特空间上算子的贝雷津数上界，利用对称均值和奥利奇函数的插值路径及改进的杨氏不等式得出结果，改进推广了已知发现并推导了不等式。
Vector-valued Reproducing Kernel Banach Spaces with Group Lasso Norms
该研究为稀疏多任务学习的核方法建立数学基础，构建了带$\ell_{p,1}$群Lasso范数的向量值再生核巴拿赫空间，证明其满足线性表示定理并提出适配的再生核，可用于稀疏多任务学习。
The paper discusses a series of results concerning reproducing kernel Hilbert spaces, related to the factorization of their kernels. In particular, it is proved that for a large class of spaces isometric multipliers are trivial. One also gives for certain spaces conditions for obtaining a particular type of dilation, as well as a classification of Brehmer type submodules.
In this paper, we study random sampling on reproducing kernel space $V$, which is a range of an idempotent integral operator. Under certain decay condition on the integral kernel, we show that any element in $V$ can be approximated by an element in a finite-dimensional subspace of $V$. Moreover, we prove with overwhelming probability that random points uniformly distributed over a cube $C$ is stable sample for the set of functions concentrated on $C$
We describe the radial Fock type spaces which possess Riesz bases of normalized reproducing kernels and which are (are not) isomorphic to de Branges spaces in terms of the weight functions.
A reproducing kernel approach to Lebesgue decomposition
该研究利用再生核理论和泛函分析，通过复单位圆上测度对的柯西变换再生核希尔伯特空间的包含与交关系，描述其控制等性质，实现勒贝格分解新构造并证明拉东-尼科迪姆定理。
By way of concrete presentations, we construct two infinite-dimensional transforms at the crossroads of Gaussian fields and reproducing kernel Hilbert spaces (RKHS), thus leading to a new infinite-dimensional Fourier transform in a general setting of Gaussian processes. Our results serve to unify existing tools from infinite-dimensional analysis.
Representing systems of reproducing kernels in spaces of analytic functions
本文用初等方法构造了哈代空间及加权哈代空间中柯西核与再生核的表示系，为相关解析函数空间的研究提供了新的构造性工具。
A close look at the entropy numbers of the unit ball of the Reproducing Hilbert Space of isotropic positive definite kernels
本文研究了正交正定核再生希尔伯特空间单位球的熵数，给出了覆盖数的精确上界和下界，并分析了其与维度和核系数的关系。
The theory of positive kernels and associated reproducing kernel Hilbert spaces, especially in the setting of holomorphic functions, has been an important tool for the last several decades in a number of areas of complex analysis and operator theory. An interesting generalization of holomorphic functions, namely free noncommutative functions (e.g., functions of square-matrix arguments of arbitrary size satisfying additional natural compatibility conditions), is now an active area of research, with motivation and applications from a variety of areas (e.g., noncommutative functional calculus, fr…
Targeting at sparse multi-task learning, we consider regularization models with an $\ell^1$ penalty on the coefficients of kernel functions. In order to provide a kernel method for this model, we construct a class of vector-valued reproducing kernel Banach spaces with the $\ell^1$ norm. The notion of multi-task admissible kernels is proposed so that the constructed spaces could have desirable properties including the crucial linear representer theorem. Such kernels are related to bounded Lebesgue constants of a kernel interpolation question. We study the Lebesgue constant of multi-task kernels…
We investigate the interaction between the existence of reproducing kernels on infinite-dimensional Hermitian vector bundles and the positivity properties of the corresponding bundles. The positivity refers to the curvature form of certain covariant derivatives associated to reproducing kernels on the vector bundles under consideration. The values of the curvature form are Hilbert space operators, and its positivity is thus understood in the usual sense from operator theory.
We give necessary and sufficient conditions under which the reproducing kernel of a Pontryagin space of $d \times 1$ vector polynomials is determined by a generalized Nevanlinna pair of $d \times d$ matrix polynomials.
Mercer's Theorem for Vector-Valued Reproducing Kernel Hilbert Spaces in Kaplansky-Hilbert Modules over $L_{\infty}(Ω)$
本文在$L_{\infty}(Ω)$上Kaplansky-希尔伯特模块上扩展了Mercer定理，通过构造等距同构和向量值提升公理化体系，建立了偏积分算子的三个基本性质的等价性。
Motivated by multi-task machine learning with Banach spaces, we propose the notion of vector-valued reproducing kernel Banach spaces (RKBS). Basic properties of the spaces and the associated reproducing kernels are investigated. We also present feature map constructions and several concrete examples of vector-valued RKBS. The theory is then applied to multi-task machine learning. Especially, the representer theorem and characterization equations for the minimizer of regularized learning schemes in vector-valued RKBS are established.
Compressions of Toeplitz operators to coinvariant subspaces of $H^2$ are called truncated Toeplitz operators. We study two questions related to these operators. The first, raised by Sarason, is whether boundedness of the operator implies the existence of a bounded symbol; the second is the reproducing kernel thesis. We show that in general the answer to the first question is negative, and we exhibit some classes of spaces for which the answers to both questions are positive.
In this paper, we mainly study the random sampling stability for signals in a weighted reproducing kernel subspace of $L_ν^p(\mathbb{R}^d)$ without the additional requirement that the kernel function has symmetry. The sampling set is independently and randomly drawn from a general probability distribution over $\mathbb{R}^d$. Based on the frame characterization of weighted reproducing kernel subspaces, we first approximate the weighted reproducing kernel space by a finite dimensional subspace on any bounded domains. Then, we prove that the random sampling stability holds with high probability …
The aim of the paper is to create a link between the theory of reproducing kernel Hilbert spaces (RKHS) and the notion of a unitary representation of a group or of a groupoid. More specifically, it is demonstrated on one hand, how to construct a positive definite kernel and an RKHS for a given unitary representation of a group(oid), and on the other hand how to retrieve the unitary representation of a group or a groupoid from a positive definite kernel defined on that group(oid) with the help of the Moore-Aronszajn theorem. The kernel constructed from the group(oid) representation is inspired …
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