论文标题
在钢化分布的不变空间中移动扩张的完整性
Completeness of shifted dilates in invariant Banach spaces of tempered distributions
论文作者
论文摘要
我们表明，从Banach模块理论和时频分析中提出的完善的方法允许在相当一般的设置中收集给定（测试）功能的移动和扩张版本的完整性结果。尽管基本思想与V. Katsnelson最近的论文中使用的论点相似，但我们将其结果扩展到多个方向，既放松了假设又扩大了应用的范围。不需要被认为嵌入$（l^2（\ Mathbb {r}），|| \ cdot {} || _2）$的Banach空间，也不相关。我们选择在欧几里得空间的设置中介绍结果，因为那时Schwartz Space $ \ Mathcal {s}^{\ prime}（\ Mathbb {r}^d）$（$ d \ geq 1 $）的恢复性分布提供了一个精心规定的数学分析环境。我们还建立了与调制空间和Shubin类$（{Q} _ {s}（\ Mathbb {r}^d），|| \ cdot {} || _ {q_s}）$的连接，表明它们是Katsnelson的特殊情况（仅），仅Katsnelson的特殊情况（仅）$ s \ geq geq 0 $。
We show that well-established methods from the theory of Banach modules and time-frequency analysis allow to derive completeness results for the collection of shifted and dilated version of a given (test) function in a quite general setting. While the basic ideas show strong similarity to the arguments used in a recent paper by V.~Katsnelson we extend his results in several directions, both relaxing the assumptions and widening the range of applications. There is no need for the Banach spaces considered to be embedded into $(L^2(\mathbb{R}), ||\cdot{}||_2)$, nor is the Hilbert space structure relevant. We choose to present the results in the setting of the Euclidean spaces, because then the Schwartz space $\mathcal{S}^{\prime}(\mathbb{R}^d)$ ($d \geq 1$) of tempered distributions provides a well-established environment for mathematical analysis. We also establish connections to modulation spaces and Shubin classes $({Q}_{s}(\mathbb{R}^d), ||\cdot{}||_{Q_s})$, showing that they are special cases of Katsnelson's setting (only) for $s \geq 0$.
