C. Bouzar, T. Saidi
The well-known characterizations of Schwartz space $\mathcal{S}$ of rapidly decreasing functions is extended to the algebra $\mathcal{G}_{\mathcal{S}}$ of rapidly decreasing generalized functions and to the algebra $\mathcal{G}_{% \mathcal{S}}^{\infty}$ of regular rapidly decreasing generalized functions.
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Let $\Mmm$ denote the set of $\mm$ matrices with complex entries, and let $\calG(\partial_1,...,\partial_n)$ be an $\mm$ matrix whose entries are partial differential operators on $\Rn$ with constant complex coefficients. It is proved that $\calG(\partial_1,...,\partial_n)\otimes δ$ is the generating distribution of a smooth one-parameter convolution semigroup of $\Mmm$-valued rapidly decreasing distributions on $\Rn$ if and only if $$\sup_{(ξ_1,...,ξ_n)\in\Rn}\hReσ(\calG(iξ_1,...,iξ_n))<\infty.$$ Applications to systems of partial differential operators with constant coefficients are consider…
Let $P(\partial_0,\partial_1,...,\partial_n)$ be a PDO on $\symR^{1+n}$ with constant coefficients. It is proved that (i) the real parts of the $λ$-roots of the polynomial $P(λ,iξ_1,...,iξ_n)$ are bounded from above when $(ξ_1,...,ξ_n)$ ranges over $\symR^n$ if and only if (ii) $P$ has a fundamental solution with support in $H_+=\{(x_0,x_1,\allowbreak..., x_n)\in \symR^{1+n}:x_0\ge0\}$ having some special properties expressed in terms of the L. Schwartz space $\calO^{\prime}_C$ of rapidly decreasing distributions. Moreover, it is proved that the fundamental solution with support in $H_+$ havin…
It is a well-known fact in K-theory that the rapidly decreasing matrices of countable size form an associative topological algebra whose set of quasi-invertible elements is open, and such that the quasi-inversion map is continuous. We generalize these conclusions to further algebras of weighted matrices with entries in a Banach algebra.
Description of linear continuous functionals on a space of rapidly decreasing infinitely differentiable functions on an unbounded closed convex set in $\mathbb R^n$ in terms of their Fourier-Laplace transform is obtained.
We use techniques from time-frequency analysis to show that the space $\mathcal S_ω$ of rapidly decreasing $ω$-ultradifferentiable functions is nuclear for every weight function $ω(t)=o(t)$ as $t$ tends to infinity. Moreover, we prove that, for a sequence $(M_p)_p$ satisfying the classical condition $(M1)$ of Komatsu, the space of Beurling type $\mathcal S_{(M_p)}$ when defined with $L^{2}\,$norms is nuclear exactly when condition $(M2)'$ of Komatsu holds.
A space of entire functions of several complex variables rapidly decreasing on ${\mathbb R}^n$ and such that their growth along $i{\mathbb R}^n$ is majorized with a help of a family of weight functions is considered in the paper. For this space an equivalent description in terms of estimates on all partial derivatives of functions on ${\mathbb R}^n$ and Paley-Wiener type theorem are obtained.
We present new types of regularity for nonlinear generalized functions, based on the notion of regular growth with respect to the regularizing parameter of Colombeau's simplified model. This generalizes the notion of G^{\infty }-regularity introduced by M. Oberguggenberger. A key point is that these regularities can be characterized, for compactly supported generalized functions, by a property of their Fourier transform. This opens the door to microanalysis of singularities of generalized functions, with respect to these regularities. We present a complete study of this topic, including proper…
We give (necessary and sufficient) conditions over a sequence $\left\{ f_{n}\right\} _{n=0}^{\infty}$ of functions under which every generalized Stieltjes moment problem \[ \int_{0}^{\infty} f_{n}(x)ϕ(x)\mathrm{d} x=a_{n}, \ \ \ n\in\mathbb{N}, \] has solutions $ϕ\in\mathcal{S}(\mathbb{R})$ with $\operatorname*{supp} ϕ\subseteq[0,\infty)$. Furthermore, we consider more general problems of this kind for measure or distribution sequences $\left\{ f_{n}\right\} _{n=0}^{\infty}$. We also study vector moment problems with values in Frechet spaces and multidimensional moment problems.
Composition and Volterra type operators on large Bergman spaces with rapidly decreasing weights
本文刻画了快速递减权大 Bergman 空间之间广义 Volterra 型积分算子的有界性、紧性与 Schatten 类性质，利用 Littlewood-Paley 公式和嵌入定理推广了 Pau 与 Peláez 的积分算子研究。
We introduce a family of differential-reflection operators $Λ_{A, \varepsilon}$ acting on smooth functions defined on $\mathbb R.$ Here $A$ is a Strum-Liouville function with additional hypotheses and $\varepsilon\in \mathbb R.$ For special pairs $(A,\varepsilon),$ we recover Dunkl's, Heckman's and Cherednik's operators (in one dimension). The spectral problem for the operators $Λ_{A, \varepsilon}$ is studied. In particular, we obtain suitable growth estimates for the eigenfunctions of $Λ_{A, \varepsilon}$. As the operators $Λ_{A, \varepsilon}$ are mixture of $d/dx$ and reflection operators, w…
We consider distributions on a closed compact manifold $M$ as maps on smoothing operators. Thus spaces of certain maps between $Ψ^{-\infty}(M)\to \mathcal{C}^{\infty}(M)$ are considered as generalized functions. For any collection of regularizing processes we produce an algebra of generalized functions and a diffeomorphism equivariant embedding of distributions into this algebra. We provide examples invariant under certain group actions. The regularity for such generalized functions is provided in terms of a certain tameness of maps between graded Frechét spaces. This notion of regularity impl…
The paper deal with the noncommutative Fréchet ${}^*$-algebra $\mathcal{L}(s',s)$ of the so-called smooth operators, i.e. linear and continuous operators acting from the space $s'$ of slowly increasing sequences to the Fréchet space $s$ of rapidly decreasing sequences. By a canonical identification, this algebra of smooth operators can be also seen as the algebra of the rapidly decreasing matrices. We give a full description of closed commutative ${}^*$-subalgebras of this algebra and we show that every closed subspace of $s$ with basis is isomorphic (as a Fréchet space) to some closed commuta…
Spectral clumping for functions decreasing rapidly on a half-line
本文证明了在半直线上快速递减的函数的Fourier变换会发生谱聚结现象，给出了非零频率聚结的条件，证明了该衰减条件接近最优，并将结果扩展到缓增分布。
Let $\mathcal{G}$ resp. $M$ be a positive dimensional Lie group resp. connected complex manifold without boundary and $V$ a finite dimensional $C^{\infty}$ compact connected manifold, possibly with boundary. Fix a smoothness class $\mathcal{F}=C^{\infty}$, Hölder $C^{k, α}$ or Sobolev $W^{k, p}$. The space $\mathcal{F}(V, \mathcal{G})$ resp. $\mathcal{F}(V, M)$ of all $\mathcal{F}$ maps $V \to \mathcal{G}$ resp. $V \to M$ is a Banach/Fréchet Lie group resp. complex manifold. Let $\mathcal{F}^0(V, \mathcal{G})$ resp. $\mathcal{F}^{0}(V, M)$ be the component of $\mathcal{F}(V, \mathcal{G})$ resp…
Let $\mathscr{L}^*(s)$ be the maximal $\mathcal{O}^*$-algebra of unbounded operators on $\ell_2$ whose domain is the space $s$ of rapidly decreasing sequences. This is a noncommutative topological algebra with involution which can be identified, for instance, with the algebra $\mathscr L(s)\cap\mathscr L(s')$ or the algebra of multipliers for the algebra $\mathscr{L}(s',s)$ of smooth compact operators. We give a simple characterization of unital commutative Fréchet ${}^*$-subalgebras of $\mathscr{L}^*(s)$ isomorphic as a Fréchet spaces to nuclear power series spaces $Λ_\infty(α)$ of infinite t…
Let s be the space of rapidly decreasing sequences. We give the spectral representation of normal elements in the Fréchet algebra L(s',s) of the so-called smooth operators. We also characterize closed commutative *-subalgebras of L(s',s) and establish a Hölder continuous functional calculus in this algebra. The key tool is the property (DN) of s.
We study composition operators on the Schwartz space of rapidly decreasing functions. We prove that such a composition operator is never a compact operator and we obtain necessary or sufficient conditions for the range of the composition operator to be closed. These conditions are expressed in terms of multipliers for the Schwartz class and the closed range property of the corresponding operator considered in the space of smooth functions.
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 …
We study the asymptotics at zero of continuous functions on (0, 1] by means of their asymptotic ideals, i.e., ideals in the ring of continuous functions on (0, 1] satisfying a polynomial growth condition at 0 modulo rapidly decreasing functions at 0. As our main result, we characterize maximal and prime ideals in terms of maximal and prime filters.
The aim of this paper is to introduce and to study the space $\mathcal{O}_{M,ω}(\mathbb{R}^N)$ of the multipliers of the space $\mathcal{S}_ω(\mathbb{R}^N)$ of the $ω$-ultradifferentiable rapidly decreasing functions of Beurling type. We determine various properties of the space $\mathcal{O}_{M,ω}(\mathbb{R}^N)$. Moreover, we define and compare some lc-topologies of which $\mathcal{O}_{M,ω}(\mathbb{R}^N)$ can be naturally endowed.
A space of entire functions of several complex variables rapidly decreasing on ${\mathbb R}^n$ and such that their growth along $i{\mathbb R}^n$ is majorized with the help of a family of weight functions is considered in this paper. For such space an equivalent description in terms of estimates on all of its partial derivatives as functions on ${\mathbb R}^n$ and a Paley-Wiener type theorem are obtained.
A new approach to the algebra G_τ of temperate nonlinear generalized functions is proposed, in which G_τ is based on the space O_{M} endowed with is natural topology in contrary to previous constructions. Thus, this construction fits perfectly in the general scheme of construction of Colombeau type algebras and reveals better properties of G_τ. This is illustrated by the natural introduction of a regularity theory in G_τ, of the Fourier transform, with the definition of G_{O_{C prime}}, the space of rapidly generalized distributions which is the Fourier image of G_τ.
We study the Schatten class membership of generalized Volterra companion integral operators on the standard Fock spaces $\mathcal{F}_α^2$. The Schatten $\mathcal{S}_p(\mathcal{F}_α^2)$ membership of the operators are characterized in terms of $L^{p/2}$ integrability of certain generalized Berezin type integral transforms on the complex plane. We also give a more simplified and easy to apply description in terms of $L^p$ integrability of the symbols inducing the operators against a supper exponentially decreasing weights. An asymptotic estimates for the $\mathcal{S}_p(\mathcal{F}_α^2)$ norms of…
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 study the spectrum of operators in the Schwartz space of rapidly decreasing functions which associate each function with its composition with a polynomial. In the case where this operator is mean ergodic we prove that its spectrum reduces to 0, while the spectrum of any non mean ergodic composition operator with a polynomial always contains the closed unit disc except perhaps the origen. We obtain a complete description of the spectrum of the composition operator with a quadratic polynomial or a cubic polynomial with positive leading coefficient.
We describe the multiplier algebra of the noncommutative Schwartz space. This multiplier algebra can be seen as the largest ${}^*$-algebra of unbounded operators on a separable Hilbert space with the classical Schwartz space of rapidly decreasing functions as the domain. We show in particular that it is neither a $\mathcal{Q}$-algebra nor $m$-convex. On the other hand, we prove that classical tools of functional analysis, for example, the closed graph theorem, the open mapping theorem or the uniform boundedness principle, are still available.
A space of entire functions of several complex variables rapidly decreasing on ${\mathbb R}^n$ and such that their growth along $i{\mathbb R}^n$ is majorized with a help of a family of weight functions (not radial in general) is considered in the paper. For this space an equivalent description in terms of estimates on all partial derivatives of functions on ${\mathbb R}^n$ and Paley-Wiener type theorem are given.
We consider nuclear function spaces on which the Weyl-Heisenberg group acts continuously and study the basic properties of the twisted convolution product of the functions with the dual space elements. The final theorem characterizes the corresponding algebra of convolution multipliers and shows that it contains all sufficiently rapidly decreasing functionals in the dual space. Consequently, we obtain a general description of the Moyal multiplier algebra of the Fourier-transformed space. The results extend the Weyl symbol calculus beyond the traditional framework of tempered distributions.
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