Claudio Carmeli, Ernesto De Vito, Alessandro Toigo
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.
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An Operator Theoretical Approach to Mercer's Theorem
本文从算子理论角度，通过再生核希尔伯特空间和紧算子谱理论，给出Mercer定理的最一般形式及其与不同分析领域之间复杂关系的现代综述。
This article studies constructions of reproducing kernel Banach spaces (RKBSs) which may be viewed as a generalization of reproducing kernel Hilbert spaces (RKHSs). A key point is to endow Banach spaces with reproducing kernels such that machine learning in RKBSs can be well-posed and of easy implementation. First we verify many advanced properties of the general RKBSs such as density, continuity, separability, implicit representation, imbedding, compactness, representer theorem for learning methods, oracle inequality, and universal approximation. Then, we develop a new concept of generalized …
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。
This is a tutorial and survey paper on kernels, kernel methods, and related fields. We start with reviewing the history of kernels in functional analysis and machine learning. Then, Mercer kernel, Hilbert and Banach spaces, Reproducing Kernel Hilbert Space (RKHS), Mercer's theorem and its proof, frequently used kernels, kernel construction from distance metric, important classes of kernels (including bounded, integrally positive definite, universal, stationary, and characteristic kernels), kernel centering and normalization, and eigenfunctions are explained in detail. Then, we introduce types …
We extend the classical Mercer theorem to reproducing kernel Hilbert spaces whose elements are functions from a measurable space $X$into $\mathbb C^n$. Given a finite measure $μ$ on $X$, we represent the reproducing kernel $K$ as convergent series in terms of the eigenfunctions of a suitable compact operator depending on $K$ and $μ$. Our result holds under the mild assumption that $K$ is measurable and the associated Hilbert space is separable. Furthermore, we show that $X$ has a natural second countable topology with respect to which the eigenfunctions are continuous and the series representi…
In this paper, we characterize the families of those bounded linear operators on a separable Hilbert space which are simultaneously unitarily equivalent to integral bi-Carleman operators on $L_2(R)$ having arbitrarily smooth kernels of Mercer type. The main result is a qualitative sharpening of an earlier result of [7].
With the aim of applications to solving general integral equations, we introduce and study in this paper a special class of bi-Carleman kernels on $\mathbb{R}\times\mathbb{R}$, called $K^\infty$ kernels of Mercer type, whose property of being infinitely smooth is stable under passage to certain left and right multiples of their associated integral operators. An expansion theorem in absolutely and uniformly convergent bilinear series concerning kernels of this class is proved extending to a general non-Hermitian setting both Mercer's and Kadota's Expansion Theorems for positive definite kernels…
In this paper, we state some characterizations of $h$-convex function is defined on a convex set in a linear space. By doing so, we extend the Jensen-Mercer inequality for $h$-convex function. We will also define $h$-convex function for operators on a Hilbert space and present the operator version of the Jensen-Mercer inequality. Lastly, we propound the complementary inequality of Jensen's inequality for $h$-convex functions.
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.
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.
Recently, there has been emerging interest in constructing reproducing kernel Banach spaces (RKBS) for applied and theoretical purposes such as machine learning, sampling reconstruction, sparse approximation and functional analysis. Existing constructions include the reflexive RKBS via a bilinear form, the semi-inner-product RKBS, the RKBS with $\ell^1$ norm, the $p$-norm RKBS via generalized Mercer kernels, etc. The definitions of RKBS and the associated reproducing kernel in those references are dependent on the construction. Moreover, relations among those constructions are unclear. We expl…
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.
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…
Spectral Theorem for Self-Adjoint Partial Integral Operators in Kaplansky-Hilbert Modules
本文在Kaplansky-希尔伯特模块中证明了自伴偏积分算子的谱定理，推广了Mercer定理并给出了具体例子。
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 …
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 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.
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.
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 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.
Representation by Integrating Reproducing Kernels
本文基于直接积分，发展了一个框架来积分参数化的再生核族，通过逐点积分得到新的再生核，其对应的Hilbert空间是各Hilbert空间直接积分在求和算子下的像，推广了有限和再生核的结果，并涵盖了Mercer核、积分变换生成核、正定函数混合及径向基函数尺度混合等特例，为Kramer采样定理等已知结果提供了新视角，并建立了测量与积分变换之间的联系。
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.
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…
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.
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
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。
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。
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}.
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.
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…
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。
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.
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。
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.
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。
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…
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。
We review definitions and properties of reproducing kernel Hilbert spaces attached to Gaussian variables and processes, with a view to applications in nonparametric Bayesian statistics using Gaussian priors. The rate of contraction of posterior distributions based on Gaussian priors can be described through a concentration function that is expressed in the reproducing Hilbert space. Absolute continuity of Gaussian measures and concentration inequalities play an important role in understanding and deriving this result. Series expansions of Gaussian variables and transformations of their reprodu…
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。
This note consists of two largely independent parts. In the first part we give conditions on the kernel $k: Ω\times Ω\rightarrow \mathbb{R}$ of a reproducing kernel Hilbert space $H$ continuously embedded via the identity mapping into $L^2(Ω, μ),$ which are equivalent to the fact that $H$ is even compactly embedded into $L^2(Ω, μ).$ In the second part we consider a scenario from infinite-variate $L^2$-approximation. Suppose that the embedding of a reproducing kernel Hilbert space of univariate functions with reproducing kernel $1+k$ into $L^2(Ω, μ)$ is compact. We provide a simple criterion fo…
We consider frames F in a given Hilbert space, and we show that every F may be obtained in a constructive way from a reproducing kernel and an orthonormal basis in an ambient Hilbert space. The construction is operator-theoretic, building on a geometric formula for the analysis operator defined from F. Our focus is on the infinite-dimensional case where a priori estimates play a central role, and we extend a number of results which are known so far only in finite dimensions. We further show how this approach may be used both in constructing useful frames in analysis and in applications, and in…
Which Spaces can be Embedded in Reproducing Kernel Hilbert Spaces?
本文研究巴拿赫函数空间嵌入再生核希尔伯特空间的条件，给出充分必要条件，并证明所需光滑性与底层维数成比例增长。
We establish the following Hilbert-space analogue of the Gleason-Kahane-Żelazko theorem. If $\mathcal{H}$ is a reproducing kernel Hilbert space with a normalized complete Pick kernel, and if $Λ$ is a linear functional on $\mathcal{H}$ such that $Λ(1)=1$ and $Λ(f)\ne0$ for all cyclic functions $f\in\mathcal{H}$, then $Λ$ is multiplicative, in the sense that $Λ(fg)=Λ(f)Λ(g)$ for all $f,g\in\mathcal{H}$ such that $fg\in\mathcal{H}$. Moreover $Λ$ is automatically continuous. We give examples to show that the theorem fails if the hypothesis of a complete Pick kernel is omitted. We also discuss cond…
Lipschitz and Hölder Continuity in Reproducing Kernel Hilbert Spaces
本文系统研究了再生核希尔伯特空间中Lipschitz与Hölder连续性的充分条件及相应再生核的刻画，并整理了已有结果，为该主题提供了统一参考。
Spectral representations of interpolation spaces of reproducing kernel Hilbert spaces
本文提出了一种对再生产核希尔伯特空间内插空间的谱表示方法，并探讨了其在统计学习中的应用。
Let T be a C_{\cdot 0}-contraction on a Hilbert space H and S be a non-trivial closed subspace of H. We prove that S is a T-invariant subspace of H if and only if there exists a Hilbert space D and a partially isometric operator Π: H^2_D(\mathbb{D}) \raro H such that ΠM_z = T Πand that S = ran Π, or equivalently, P_S = ΠΠ^*. As an application we completely classify the shift-invariant subspaces of C_{\cdot 0}-contractive and analytic reproducing kernel Hilbert spaces over the unit disc. Our results also includes the case of weighted Bergman spaces over the unit disk.
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 …
Kernel methods, being supported by a well-developed theory and coming with efficient algorithms, are among the most popular and successful machine learning techniques. From a mathematical point of view, these methods rest on the concept of kernels and function spaces generated by kernels, so called reproducing kernel Hilbert spaces. Motivated by recent developments of learning approaches in the context of interacting particle systems, we investigate kernel methods acting on data with many measurement variables. We show the rigorous mean field limit of kernels and provide a detailed analysis of…
Chains of reproducing kernel Hilbert spaces generated by unimodular functions
本文利用满足条件的幺模函数与一阶微分方程组构造再生核 Hilbert 空间链，并据此给出从 de Branges 空间恢复结构 Hamiltonian 的条件性一般反问题解法。
A Schur-class function in $d$ variables is defined to be an analytic contractive-operator valued function on the unit polydisk. Such a function is said to be in the Schur--Agler class if it is contractive when evaluated on any commutative $d$-tuple of strict contractions on a Hilbert space. It is known that the Schur--Agler class is a strictly proper subclass of the Schur class if the number of variables $d$ is more than two. The Schur--Agler class is also characterized as those functions arising as the transfer function of a certain type (Givone--Roesser) of conservative multidimensional line…
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