论文标题
希尔伯特空间的链条和尺度中的框架相关序列
Frame-related Sequences in Chains and Scales of Hilbert Spaces
论文作者
论文摘要
希尔伯特空间的框架对于数学家来说很有趣，但对于应用程序也很重要，例如在信号分析和物理学中。在数学和物理学中，自然要考虑一个完整的空间，而不仅仅是一个空间。在本文中，我们研究了某些与框架相关的属性是如何作为在一个空间中某个序列的（半）框架的完整性或（半）框架的特性，以量表传播到其他空间。我们将其链接到各个与框架相关的操作员的属性，例如分析或合成。我们从对希尔伯特连锁店理论的详细调查开始。使用规范的同构，框架序列的特性自然保留在不同空间之间。我们还表明，如果考虑原始序列，可以转移一些结果，特别是将上半框架的属性保存在较大的空间中，而下部的属性则保持在较小的空间中。这导致了负面的结果：如果尺度不平凡，即空间不相等，则序列永远不可能成为量表的两个希尔伯特空间的框架。
Frames for Hilbert spaces are interesting for mathematicians but also important for applications e.g. in signal analysis and in physics. Both in mathematics and physics it is natural to consider a full scale of spaces, and not only a single one. In this paper, we study how certain frame-related properties, as completeness or the property of being a (semi-)frame, of a certain sequence in one of the spaces propagate to other spaces in a scale. We link that to the properties of the respective frame-related operators, like analysis or synthesis. We start with a detailed survey of the theory of Hilbert chains. Using a canonical isomorphism the properties of frame sequences are naturally preserved between different spaces. We also show that some results can be transferred if the original sequence is considered, in particular that the upper semi-frame property is kept in larger spaces, while the lower one to smaller ones. This leads to a negative result: a sequence can never be a frame for two Hilbert spaces of the scale if the scale is non-trivial, i.e. spaces are not equal.
