The unitary extension principle for locally compact abelian groups with co-compact subgroups,Proceedings of the American Mathematical Society

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The unitary extension principle for locally compact abelian groups with co-compact subgroups
Proceedings of the American Mathematical Society ( IF 1 ) Pub Date : 2021-01-22 , DOI: 10.1090/proc/15319
Ole Christensen , Say Song Goh

Abstract:The unitary extension principle by Ron and Shen is one of the cornerstones of wavelet frame theory; it leads to tight frames for $L^{2}(\mathbb{R})$ and associated expansions of functions $f\in L^{2}(\mathbb{R})$ of similar type as those for orthonormal wavelet bases. In this paper, the unitary extension principle is extended to the setting of a locally compact abelian group, equipped with a collection of nested co-compact subgroups. Unlike all previously known generalizations of the unitary extension principle, the current one is taking place within the setting of continuous frames, which means that the resulting decompositions of functions in the underlying Hilbert space in general are given in terms of integral representations rather than discrete sums. The frame elements themselves appear by letting a collection of modulation operators act on a countable family of basic functions.

中文翻译：

具有局部紧群的亚紧群的单一扩张性原理

【摘要】罗恩和申的单一扩张原理是小波框架理论的基石之一。导致框架紧缩以及相关的功能扩展 $L ^ {2}（\ mathbb {R}）$ $f \ in L ^ {2}（\ mathbb {R}）$ 与正交小波基的类型相似。在本文中，将一元可扩展原理扩展到局部紧致的阿贝尔群的设置，该群配备了嵌套的紧致子群的集合。不同于所有先前已知的extension扩展原理的概括，当前的展开是在连续帧的设置内进行的，这意味着基础希尔伯特空间中的函数分解结果通常以积分表示而不是离散和给出。。通过让一组调制算子作用于可数的基本功能族上，可以显示帧元素本身。

更新日期：2021-02-08

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