A convenient technique for proving kernel theorems for (LF)-spaces (countable inductive limits of Frechet spaces)is developed. The proposed approach is based on introducing a suitable modification of the functor of the completed inductive topological tensor product. Using such modified tensor products makes it possible to prove kernel theorems without assuming the completeness of the considered (LF)-spaces. The general construction is applied to proving kernel theorems for a class of spaces of entire analytic functions arising in nonlocal quantum field theory.
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.
We prove new theorems about properties of generalized functions defined on Gelfand-Shilov spaces $S^β$ with $0\leβ<1$. For each open cone $U\subset\mathbb R^d$ we define a space $S^β(U)$ which is related to $S^β(\mathbb R^d)$ and consists of entire analytic functions rapidly decreasing inside U and having order of growth $\le 1/(1-β)$ outside the cone. Such sheaves of spaces arise naturally in nonlocal quantum field theory, and this motivates our investigation. We prove that the spaces $S^β(U)$ are complete and nuclear and establish a decomposition theorem which implies that every continuous f…
We prove general kernel theorems for operators acting between coorbit spaces. These are Banach spaces associated to an integrable representation of a locally compact group and contain most of the usual function spaces (Besov spaces, modulation spaces, etc.). A kernel theorem describes the form of every bounded operator between a coorbit space of test functions and distributions by means of a kernel in a coorbit space associated to the tensor product representation. As special cases we recover Feichtinger's kernel theorem for modulation spaces and the recent generalizations by Cordero and Nicol…
A convenient technique for calculating completed topological tensor products of functional Frechet or DF spaces is developed. The general construction is applied to proving kernel theorems for a wide class of spaces of smooth and entire analytic functions.
We study the expansions of the elements in $\mathcal S(\mathbb{R}_+^d)$ and $\mathcal{S}'(\mathbb{R}_+^d)$ with respect to the Laguerre orthonormal basis, extending the result of M. Guillemont-Teissier in the case $d=1$. As a consequence, we obtain the Schwartz kernel theorem for $\mathcal{S}(\mathbb{R}_+^d)$ and $\mathcal{S}'(\mathbb{R}_+^d)$ and the extension theorem of Whitney type for $\mathcal{S}(\mathbb{R}_+^d)$.
In connection with the classical Schwartz kernel theorem, we show that in the framework of Colombeau generalized functions a large class of linear mappings admit integral kernels. To do this, we need to introduce news spaces of generalized functions with slow growth and the corresponding adapted linear mappings. Finally, we show that in some sense Schwartz' result is contained in our main theorem.
In the framework of idempotent mathematics, analogs of the classical kernel theorems of L. Schwartz and A. Grothendieck are studied. Idempotent versions of nuclear spaces (in the sense of A. Grothendieck) are discussed. The so-called algebraic approach is used. This means that the basic concepts and results (including those of "topological" nature) are simulated in purely algebraic terms.
We give a simple proof of the Kernel theorem for the space of tempered ultradistributions of Beurling - Komatsu type, using the characterization of Fourier-Hermite coefficients of the elements of the space. We prove in details that the test space of tempered ultradistributions of Beurling - Komatsu type can be identified with the space of sequences of ultrapolynomal falloff and its dual space with the space of sequences of ultrapolynomial growth. As a consequence of the Kernel theorem we have that the Weyl transform can be extended on a space of tempered ultradistributions of Beurling - Komats…
It was shown recently that the space isomorphic with an Gelfand Shilov space is well adapted for the use in quantum field theory with a fundamental length. It is our believe that all Gelfand Shilov spaces, especially those with quasianalytic test function spaces, are good domains for the quantum field theory. The theory requires technical results from the theory of generalized functions and not merely differential calculus and well defined Fourier transform, but also the kernel theorem and the structural theorem. In the paper we give the structural (regularity) theorem and kernel theorem for G…
We prove a new dyadic representation theorem with applications to the $T(1)$ and $A_2$ theorems. In particular, we obtain the non-homogeneous $T(1)$ theorem under weaker kernel regularity than the earlier approaches.
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 …
In this note we describe conditions under which, in idempotent functional analysis, linear operators have integral representations in terms of idempotent integral of V. P. Maslov. We define the notion of nuclear idempotent semimodule and describe idempotent analogs of the classical kernel theorems of L. Schwartz and A. Grothendieck. Our results provide a general description of a class of subsemimodules of the semimodule of all bounded functions with values in the Max-Plus algebra where some kind of kernel theorem holds, thus addressing an open problem posed by J. Gunawardena. Previously, some …
Let G be a solvable Lie group endowed with right Haar measure. We define and study a dense Frechet *-subalgebra S of L1(G), consisting of smooth functions rapidly-decreasing at infinity on G. When G is nilpotent, we recover the classical Schwartz algebra introduced by R. Howe and other authors. We develop a distribution theory for S, and we generalize the classical Kernel Theorem of L. Schwartz to our setting.
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…
The first main result of this note, Theorem 1.2, establishes the determinantal identities (7) and (8) for the expectation, under a determinantal point process governed by an integrable projection kernel, of scaling limits of characteristic polynomials sampled at several points. The determinantal identities (7) and (8) can be seen as the scaling limit of the identity of Fyodorov and Strahov for the averages of ratios of products of the values of the characteristic polynomial of a Gaussian unitary matrix. Borodin, Olshanski and Strahov derived the determinantal identity of Fyodorov and Strahov f…
We prove some inequalities for the spectral radius of positive operators on Banach function spaces. In particular, we show the following extension of Levinger's theorem. Let $K$ be a positive compact kernel operator on $L^2(X,μ)$ with the spectral radius $r(K)$. Then the function $ϕ$ defined by $ϕ(t) = r(t K + (1-t) K^*)$ is non-decreasing on $[0, {1/2}]$. We also prove that $\| A + B^* \| \ge 2 \cdot \sqrt{r(A B)}$ for any positive operators $A$ and $B$ on $L^2(X,μ)$.
The Segal algebra $\mathbf{S}_{0}(G)$ is well defined for arbitrary locally compact Abelian Hausdorff (LCA) groups $G$. It is a Banach space that exhibits a kernel theorem similar to the well-known Schwartz kernel theorem. Specifically, we call this characterization of the continuous linear operators from $\mathbf{S}_{0}(G_{1})$ to $\mathbf{S}'_{0}(G_{2})$ by generalized functions in $\mathbf{S}'_{0}(G_{1} \times G_{2})$ the "outer kernel theorem". The main subject of this paper is to formulate what we call the "inner kernel theorem". This is the characterization of those linear operators that…
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 analogy to the classical isomorphism between $\mathcal{L}(\mathcal{S}(\mathbb{R}^{n}) ,\mathcal{S}^{\prime}(\mathbb{R}^{m}) ) $ and $\mathcal{S}^{\prime}(\mathbb{R}^{n+m}) $, we show that a large class of moderate linear mappings acting between the space $\mathcal{G}\_{\mathcal{S}}(\mathbb{R}^{n}) $ of Colombeau rapidly decreasing generalized functions and the space $\mathcal{G}\_τ(\mathbb{R}^{n}) $ of temperate ones admits generalized integral representations, with kernels belonging to $\mathcal{G}\_τ(\mathbb{R}^{n+m}) $. Furthermore, this result contains the classical one in the sense of …
The main result of this paper, Theorem 1.5, establishes a conjecture of Lyons and Peres: for a determinantal point process governed by a reproducing kernel, the system of kernels sampled at the particles of a random configuration is complete in the range of the kernel. A key step in the proof, Lemma 1.11, states that conditioning on the configuration in a subset preserves the determinantal property, and the main Lemma 1.12 is a new local property for kernels of conditional point processes. In Theorem 1.7 we prove the triviality of the tail sigma-algebra for determinantal point processes govern…
We study the nuclearity of the Gelfand-Shilov spaces $\mathcal{S}^{(\mathfrak{M})}_{(\mathscr{W})}$ and $\mathcal{S}^{\{\mathfrak{M}\}}_{\{\mathscr{W}\}}$, defined via a weight (multi-)sequence system $\mathfrak{M}$ and a weight function system $\mathscr{W}$. We obtain characterizations of nuclearity for these function spaces that are counterparts of those for Köthe sequence spaces. As an application, we prove new kernel theorems. Our general framework allows for a unified treatment of the Gelfand-Shilov spaces $\mathcal{S}^{(M)}_{(A)}$ and $\mathcal{S}^{\{M\}}_{\{A\}}$ (defined via weight seq…
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.
We give a modified definition of a reproducing kernel Hilbert $C^*$-module (shortly, $RKHC^*M$) without using the condition of self-duality and discuss some related aspects; in particular, an interpolation theorem is presented. We investigate the exterior tensor product of $RKHC^*M$s and find their reproducing kernel. In addition, we deal with left multipliers of $RKHC^*M$s. Under some mild conditions, it is shown that one can make a new $RKHC^*M$ via a left multiplier. Moreover, we introduce the Berezin transform of an operator in the context of $RKHC^*M$s and construct a unital subalgebra of…
We consider positive semidefinite kernels valued in the $*$-algebra of adjointable operators on a VE-space (Vector Euclidean space) and that are invariant under actions of $*$-semigroups. A rather general dilation theorem is stated and proved: for these kind of kernels, representations of the $*$-semigroup on either the VE-spaces of linearisation of the kernels or on their reproducing kernel VE-spaces are obtainable. We point out the reproducing kernel fabric of dilation theory and we show that the general theorem unifies many dilation results at the non topological level.
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.
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…
Schur-Horn theorems focus on determining the diagonal sequences obtainable for an operator under all possible basis changes, formally described as the range of the canonical conditional expectation of its unitary orbit. Following a brief background survey, we prove an infinite dimensional Schur-Horn theorem for positive compact operators with infinite dimensional kernel, one of the two open cases posed recently by Kaftal- Weiss. There, they characterized the diagonals of operators in the unitary orbits for finite rank or zero kernel positive compact operators. Here we show how the characteriza…
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.
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.
Hilbertian kernel methods and their positive semidefinite kernels have been extensively used in various fields of applied mathematics and machine learning, owing to their several equivalent characterizations. We here unveil an analogy with concepts from tropical geometry, proving that tropical positive semidefinite kernels are also endowed with equivalent viewpoints, stemming from Fenchel-Moreau conjugations. This tropical analogue of Aronszajn's theorem shows that these kernels correspond to a feature map, define monotonous operators, and generate max-plus function spaces endowed with a repro…
We provide a short argument to establish a Beurling-Lax-Halmos theorem for reproducing kernel Hilbert spaces whose kernel has a complete Nevanlinna-Pick factor. We also record factorization results for pairs of nested invariant subspaces.
We prove that for every radial weighted Fock space, the system biorthogonal to a complete and minimal system of reproducing kernels is also complete under very mild regularity assumptions on the weight. This result generalizes a theorem by Young on reproducing kernels in the Paley--Wiener space and a recent result of Belov for the classical Bargmann--Segal--Fock space.
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.
For a constant coefficient partial differential operator $P(D)$ with a single characteristic direction such as the time-dependent free Schrödinger operator as well as non-degenerate parabolic differential operators like the heat operator we characterize when open subsets $X_1\subseteq X_2$ of $\mathbb{R}^d$ form a $P$-Runge pair. The presented condition does not require any kind of regularity of the boundaries of $X_1$ nor $X_2$. As part of our result we prove that for a large class of non-elliptic operators $P(D)$ there are smooth solutions $u$ to the equation $P(D)u=0$ on $\mathbb{R}^d$ with…
The purpose of this paper is to construct a bivariate generalization of new family of Kantorovich type sampling operators $(K_w^φf)_{w>0}.$ First, we give the pointwise convergence theorem and a Voronovskaja type theorem for these Kantorovich generalized sampling series. Further, we obtain the degree of approximation by means of modulus of continuity and quantitative version of Voronovskaja type theorem for the family $(K_w^φf)_{w>0}.$ Finally, we give some examples of kernels such as box spline kernels and Bochner-Riesz kernel to which the theory can be applied.
We consider a class of maps from integral Hankel operators to Hankel matrices, which we call restriction maps. In the simplest case, such a map is simply a restriction of the integral kernel onto integers. More generally, it is given by an averaging of the kernel with a sufficiently regular weight function. We study the boundedness of restriction maps with respect to the operator norm and the Schatten norms.
An abstract Pick interpolation theorem for a family of positive semi-definite kernels on a set $X$ is formulated. The result complements those in \cite{Ag} and \cite{AMbook} and will subsequently be applied to Pick interpolation on distinguished varieties \cite{JKM}.
In this note, we give a new property of de Branges-Rovnyak kernels. As the main theorem, it is shown that the exponential of de Branges-Rovnyak kernel is strictly positive definite if the inner part of the corresponding Schur class function is nontrivial.
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.
