论文标题
通过Tauberian条件不变的Banach空间中不变的Banach空间中的一组班次的完整性
Completeness of Sets of Shifts in Invariant Banach Spaces of Tempered Distributions via Tauberian conditions
论文作者
论文摘要
本文的主要结果是V. Katsnelson在最近的一篇论文[35]中对完整结果的概括远远达到了概括。与其仅使用一系列扩张的高斯人，还表明作者是早期论文[27]的关键步骤，结合使用Tauberian条件（即傅立叶变换的不变），使我们可以证明单个功能$ g \ in {\ BoldSymbol {\ Boldsymbol {\ Boldsymbol {\ Boldsymbol {\ Boldsymbol {\ boldsymbol { {s}（{\ MathBb {r}}^d）}} $是满足某些双重不变属性的任何Banach空间的密集子空间。 实际上，提出了一个更强有力的陈述：对于给定的紧凑型子集$ m $，在这种班克空间$（{\ boldsymbol b}，\，\，\ | \，\ cdot \，\ | _ {\ boldsymbol b}）$都可以构建有限排名的运算范围内，该范围内的范围$ g $ g $ g $ g $ g $ g $ g， $ m $达到给定的精度。 钢化分布的设置允许将技术参数减少为傅立叶分析中广泛使用的方法。非Quasi分析权重的扩展分别将局部紧凑的Abelian群体留给即将到来的纸张，从技术上讲，该论文将更加参与其中并使用不同的成分。
The main result of this paper is a far reaching generalization of the completeness result given by V.~Katsnelson in a recent paper [35]. Instead of just using a collection of dilated Gaussians it is shown that the key steps of an earlier paper [27] by the authors, combined with the use of Tauberian conditions (i.e. the non-vanishing of the Fourier transform) allow us to show that the linear span of the translates of a single function $g \in {\boldsymbol{\mathcal {S}({\mathbb{R}}^d)}}$ is a dense subspace of any Banach space satisfying certain double invariance properties. In fact, a much stronger statement is presented: for a given compact subset $M$ in such a Banach space $({\boldsymbol B}, \, \|\,\cdot\,\|_{\boldsymbol B})$ one can construct a finite rank operator, whose range is contained in the linear span of finitely many translates of $g$, and which approximates the identity operator over $M$ up to a given level of precision. The setting of tempered distributions allows to reduce the technical arguments to methods which are widely used in Fourier Analysis. The extension to non-quasi-analytic weights respectively locally compact Abelian groups is left to a forthcoming paper, which will be technically much more involved and uses different ingredients.
