The Density Theorem for Operator-Valued Frames via Square-Integrable Representations of Locally Compact Groups

IF 16.4 1区 化学 Q1 CHEMISTRY, MULTIDISCIPLINARY
Jingsheng Wang, Pengtong Li, Deguang Han
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引用次数: 0

Abstract

In this paper, we first prove a density theorem for operator-valued frames via square-integrable representations restricted to closed subgroups of locally compact groups, which is a natural extension of the density theorem in classical Gabor analysis. More precisely, it is proved that for such an operator-valued frame, the index subgroup is co-compact if and only if the generator is a Hilbert–Schmidt operator. Then we present some applications of this density theorem, and in particular establish necessary and sufficient conditions for the existence of such operator-valued frames with Hilbert–Schmidt generators. We also introduce the concept of wavelet transform for Hilbert–Schmidt operators, and use it to prove that if the representation space is infinite-dimensional, then the system indexed by the entire group is Bessel system but not a frame for the space of all Hilbert–Schmidt operators on the representation space.

通过局部紧凑群的平方不可穷举表示的算子值帧密度定理
在本文中,我们首先通过限制在局部紧凑群的闭合子群中的方整表示,证明了算子值帧的密度定理,这是对经典 Gabor 分析中的密度定理的自然扩展。更确切地说,我们证明了对于这样的算子值框架,只有当且仅当生成器是希尔伯特-施密特算子时,索引子群才是共紧密的。然后,我们介绍了这一密度定理的一些应用,特别是为这种具有希尔伯特-施密特生成器的算子值框架的存在建立了必要条件和充分条件。我们还引入了希尔伯特-施密特算子的小波变换概念,并用它证明,如果表示空间是无穷维的,那么以整个群为索引的系统是贝塞尔系统,但不是表示空间上所有希尔伯特-施密特算子空间的框架。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Accounts of Chemical Research
Accounts of Chemical Research 化学-化学综合
CiteScore
31.40
自引率
1.10%
发文量
312
审稿时长
2 months
期刊介绍: Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.
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