A. G. Smirnov, M. A. Soloviev
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.
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In this paper, we prove the Smulian s theorem on Frechet differentiability of norm,and present some of its geometric results concerning the Gateaux and Frechet differentiability of norm and properties of the allied space and its dual such as reflexivity and strict convexity.
We prove an inverse function theorem of Nash-Moser type for maps between Fréchet spaces satisfying tame estimates. In contrast to earlier proofs, we do not use the Newton method, that is, we do not use quadratic convergence to overcome the lack of derivatives. In fact, our theorem holds when the map to be inverted is not C^2
I present an inverse function theorem for differentiable maps between Frechet spaces which contains the classical theorem of Nash and Moser as a particular case. In contrast to the latter, the proof does not rely on the Newton iteration procedure, but on Lebesgue's dominated convergence theorem and Ekeland's variational principle. As a consequence, the assumptions are substantially weakened: the map F to be inverted is not required to be C^2, or even C^1, or even Frechet-differentiable.
In this paper we consider Bernstein's Lethargy Theorem (BLT) in the context of Fréchet spaces. Let $X$ be an infinite-dimensional Fréchet space and let $\mathcal{V}=\{V_n\}$ be a nested sequence of subspaces of $ X$ such that $ \bar{V_n} \subseteq V_{n+1}$ for any $ n \in \mathbb{N}$ and $ X=\bar{\bigcup_{n=1}^{\infty}V_n}.$ Let $ e_n$ be a decreasing sequence of positive numbers tending to 0. Under an additional natural condition on $\sup\{\{dist}(x, V_n)\}$, we prove that there exists $ x \in X$ and $ n_o \in \mathbb{N}$ such that $$ \frac{e_n}{3} \leq \{dist}(x,V_n) \leq 3 e_n $$ for any $ …
A convenient technique for calculating completed topological tensor products of functional Frechet and 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.
The aim of this paper is to to show the admissibility of some class of Frechet spaces (see Definition 2.3). In particular, this generalizes the main results of [3]. As an application, we show the admissibility of a large class modular spaces equipped with F-norms determined in Theorem 4.1. It is worth noticing that F-norms introduced in Theorem 4.1 generalize the classical Luxemburg F-norm. Also a linear version of admissibility (so called metric approximation property) for order continuous symmetric spaces will be demonstrated (see Theorem 5.1).
Inverse Function Theorem in Fréchet Spaces
本文利用Ekeland等人的近期进展和变分分析思想，在Fréchet空间中建立了Nash–Moser型反函数定理的一般满射性与Lipschitz连续性结果，并应用于微分方程。
We prove an implicit function theorem for Keller C^k_c-maps from arbitrary real or complex topological vector spaces to Frechet spaces, imposing only a certain metric estimate on the partial differentials. As a tool, we show the C^k-dependence of fixed points on parameters for suitable families of contractions of a Frechet space. The investigations were stimulated by a recent metric approach to differentiability in Frechet spaces by Olaf Mueller. Our results also subsume generalizations of Mueller's Inverse Function Theorem for mappings between Frechet spaces. As an application, we prove exist…
We give a Hahn-Jordan decomposition in Riesz spaces which generalizes that of [{{\sc B. A. Watson}, {An Andô-Douglas type theorem in Riesz spaces with a conditional expectation,} {\em Positivity,} {\bf 13} (2009), 543 - 558}] and a Riesz-Frechet representation theorem for the $T$-strong dual, where $T$ is a Riesz space conditional expectation operator. The result of Watson was formulated specifically to assist in the proof of the existence of Riesz space conditional expectation operators with given range space, i.e., a result of Andô-Douglas type. This was needed in the study of Markov process…
Fixed sets and fixed points in $\Lim$--spaces]{Fixed sets and fixed points for mappings in generalized $\Lim$--spaces of Fréchet
本文在公理化定义的广义Fréchet $\Lim$--空间中，借助偏序集值类距离函数研究压缩映射，并建立了Ćirić型和Caristi型广义压缩的不动集与不动点定理。
A theorem of Godefroy and Shapiro states that non-trivial convolution operators on the space of entire functions on $\mathbb{C}^n$ are hypercyclic. Moreover, it was shown by Bonilla and Grosse-Erdmann that they have frequently hypercyclic functions of exponential growth. On the other hand, in the infinite dimensional setting, the Godefroy-Shapiro theorem has been extended to several spaces of entire functions defined on Banach spaces. We prove that on all these spaces, non-trivial convolution operators are strongly mixing with respect to a gaussian probability measure of full support. For the …
Given work contains the full text of the proof of the following assertion: For the topological algebra $C^{\infty}(\mathcal{M})$ of smooth functions on a smooth $m$-dimensional real manifold $\mathcal{M}$ the small global dimension $(\mathop{\mathrm{ds}} C^\infty (\mathcal{M}))$, the global homological dimension $(\mathop{\mathrm{dg}} C^\infty (\mathcal{M}))$ and the bidimension $(\mathop{\mathrm{db}} C^\infty(\mathcal{M}))$ are equal to $m$ (all dimensions are understood in the sense of the homology of topological (locally convex) algebras).
When dealing with concrete problems in a function space on R^n, it is sometimes helpful to have a dense subspace consisting of functions of a particular type, adapted to the problem under consideration. We give a theorem that allows one to write down many of such subspaces in commonly occurring Fréchet function spaces. These subspaces are all of the form $\{pf_0 | p\in{\cal P}\}$ where $f_0$ is a fixed function and ${\cal P}$ is an algebra of functions. Classical results like the Stone-Weierstrass theorem for polynomials and the completeness of the Hermite functions are related by this theorem…
In this paper, we give conditions under which we can compute the conjugate of a convex function on the product of two Frechet spaces defined in terms of another convex function on the product of two (possibly different) Frechet spaces. We use this result to give simple proofs of some (both old and new) results for Banach spaces, and deduce some (both old and new) stability results for strongly representable multifunctions. We take as our starting point a result on closed convex cones in the product of two Frechet spaces.
Frechet and Mordukhovich Derivative (Coderivative) and Covering Constant for Single-Valued Mapping in Euclidean Space with Applications (I)
本文研究欧几里得空间中单值映射的Frechet与Mordukhovich导数计算规则，并利用所得导数解求覆盖常数，进而结合参数化重合点定理求解参数化方程。
The first Fundamental Theorem of Calculus for functions defined on Wasserstein space
本文在Wasserstein空间上建立了微积分基本定理的类比，证明了足够规则的函数积分可微且导数与被积函数一致。
It is well known, as follows from the Banach-Steinhaus theorem, that if a sequence $\left\{y_{n}\right\}_{n=1}^{\infty}$ of linear continuous functionals in a Fréchet space converges pointwise to a linear functional $Y,$ $Y\left( x\right) =\lim_{n\rightarrow\infty}\left\langle y_{n},x\right\rangle $ for all $x,$ then $Y$ is actually continuous. In this article we prove that in a Fréchet space the continuity of $Y$ still holds if $Y$ is the \emph{finite part} of the limit of $\left\langle y_{n},x\right\rangle $ as $n\rightarrow\infty.$ We also show that the continuity of finite part limits hold…
Frechet and Mordukhovich Derivative (Coderivative) and Covering Constant for Single-Valued Mapping in Euclidean Space with Application (II)
本文针对欧几里得空间中的保范单值映射，计算其Fréchet导数与Mordukhovich导数，证明除原点外覆盖常数恰为1，并将结果推广到R^4后用于求解参数化方程。
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 prove a Nash-Moser type inverse function theorem in Frechet spaces for functions with approximate inverses, allowing for a loss of derivatives proportional to $n$ in the way of Lojasiewicz and Zehnder.
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.
Linear topological invariants for kernels of differential operators by shifted fundamental solutions
本文利用满足特定性质的移位基本解，刻画了偏微分算子光滑核与分布核的线性拓扑条件$(Ω)$、$(PΩ)$及$(P\overline{\overlineΩ})$，并基于空间提升定理给出微分方程解参数依赖问题的特征与新证明。
For the Frechet space E=C^{\infty}(S^1) and for a smooth ϕ: R to R, we prove that the associated map E to E given by x mapstoϕ\circ x satisfies the continuous BΓ--differentiability condition in Yamamuro's inverse function theorem only if ϕis affine. Via more complicated examples, we also generally discuss the importance of testing the applicability of proposed inverse and implicit function theorems by this kind of simple maps.
In this note we prove a Birkhoff type transitivity theorem for continuous maps acting on non-separable completely metrizable spaces and we give some applications for dynamics of bounded linear operators acting on complex Fréchet spaces. Among them we show that any positive power and any unimodular multiple of a topologically transitive linear operator is topologically transitive, generalizing similar results of S.I. Ansari and F. León-Saavedra V. Müller for hypercyclic operators.
Chaotic linear dynamics deals primarily with various topological ergodic properties of semigroups of continuous linear operators acting on a topological vector space. We treat questions of characterizing which of the spaces from a given class support a semigroup of prescribed shape satisfying a given topological ergodic property. In particular, we characterize countable inductive limits of separable Banach spaces that admit a hypercyclic operator, show that there is a non-mixing hypercyclic operator on a separable infinite dimensional complex Fréchet space $X$ if and only if $X$ is non-isomorp…
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…
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.
Let $K$ be a Hausdorff space and $C_b(K)$ be the Banach algebra of all complex bounded continuous functions on $K$. We study the Gâteaux and Fréchet differentiability of subspaces of $C_b(K)$. Using this, we show that the set of all strong peak functions in a nontrivial separating separable subspace $H$ of $C_b(K)$ is a dense $G_δ$ subset of $H$, if $K$ is compact. This gives a generalized Bishop's theorem, which says that the closure of the set of strong peak point for $H$ is the smallest closed norming subset of $H$. The classical Bishop's theorem was proved for a separating subalgebra $H$ a…
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.
From Riesz to Kakutani: Representation Theorems and the Analytical Foundations of Probablility
本文通过从里斯到卡库塔尼的表示定理，揭示概率论作为泛函分析的几何实现，展现期望、分布等概念在函数空间中的存在性和唯一性原则。
Kernel theorems for Beurling-Björck type spaces
本文针对由权函数ω和η定义的一般Beurling-Björck型空间，证明了新的核定理，特别覆盖了经典Beurling-Björck空间。
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…
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.
Following [2], a Tychonoff space $X$ is Ascoli if every compact subset of $C_k(X)$ is equicontinuous. By the classical Ascoli theorem every $k$-space is Ascoli. We show that a strict $(LF)$-space $E$ is Ascoli iff $E$ is a Fréchet space or $E=ϕ$. We prove that the strong dual $E'_β$ of a Montel strict $(LF)$-space $E$ is an Ascoli space iff one of the following assertions holds: (i) $E$ is a Fréchet--Montel space, so $E'_β$ is a sequential non-Fréchet--Urysohn space, or (ii) $E=ϕ$, so $E'_β= \mathbb{R}^ω$. Consequently, the space $\mathcal{D}(Ω)$ of test functions and the space of distribution…
We prove Beurling's theorem for rank 1 Riemmanian symmetric spaces and relate it to the characterization of the heat kernel of the symmetric space.
The Mercer-Young Theorem for Matrix-Valued Kernels on Separable Metric Spaces
将 Mercer-Young 正定核刻画定理推广至可分离度量空间上的矩阵值核，并展示其在凸优化等领域的应用。
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 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…
Banach Gelfand Triples with Kernel Theorems via Equivalence Bimodules
本文通过等价双模构造Banach Gelfand三元组，证明有限生成投影Hilbert C*-模的核定理，并应用于Heisenberg模，得到时频分析的新核定理。
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.
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 …
On a space of functions with entire Laplace transforms and its connection with the optimality of the Ingham-Karamata theorem
本文研究一类 Laplace 变换可整延拓的连续可微函数所构成的 Fréchet 空间的逼近性质，并利用其与开映射定理证明 Ingham–Karamata Tauberian 定理的最优性。
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