Topological algebras of rapidly decreasing matrices and generalizations

One-parameter convolution semigroups of rapidly decreasing distributions

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…

A new study on the absolute matrix summability of non-decreasing sequences

Recently, in [Bor4], Bor proved a main theorem dealing with $|\bar{N}, p_{n}|_{k}$ summability factors of infinite series. In the present paper, we have generalized that theorem for $|A, p_{n}|_{k}$ summability method by taking normal matrices in place of weighted mean matrices.

Fundamental solutions of evolutionary PDOs and rapidly decreasing distributions

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…

General Stieltjes moment problems for rapidly decreasing smooth functions

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.

Ultrarapidly decreasing ultradifferentiable functions, Wigner distributions and density matrices

Spaces $S_ω, S_{\{ω\}}, S_{(ω)}$ of ultradecreasing ultradifferentiable (or for short, ultra-S) functions, depending on a weight $e^{ω(x)}$, are introduced in the context of quantum statistics. The corresponding coefficient spaces in the Fock basis are identified, and it is shown that the Hermite expansion is a tame isomorphism between these spaces. These results are used to link decrease properties of density matrices to corresponding properties of the Wigner distribution.

关于具有快速递减权的大 Bergman 空间上的复合算子与 Volterra 型算子

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。

On a family of differential-reflection operators: intertwining operators and Fourier transform of rapidly decreasing functions

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…

Fréchet algebras with a dominating Hilbert algebra norm

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…

On the algebra of smooth operators

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.

在半直线上快速递减的函数的谱聚结

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。

Wiener型定理对于可数极限的准Beurling代数及权重的最大化结果

Wiener type theorems for countable limits of quasi-Beurling algebras and maximizing results on weights

本文研究了可数极限的准Beurling代数的Wiener型定理，并通过权重分析得到逆闭代数的层次结构及非可接受加权Wiener定理的权重最大化结果。