Multiresolution analysis is a mathematical tool used to decompose functions in different resolution subspaces, where the scaling function plays a key role to construct the nested subspaces in \(L^{2}({\mathbb {R}})\). This paper presents a generalization of the same in \(L^{2}({\mathbb {Q}}_p)\), called frame multiresolution analysis (FMRA). So FMRA is a generalization of multiresolution analysis with frame condition. We study various properties of FMRA including characterizations in \(L^{2}({\mathbb {Q}}_p)\). Furthermore, frame scaling sets are studied with examples.

中文翻译：

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$${\mathbb {Q}}_p$$ 上的框架多分辨率分析

多分辨率分析是一种数学工具，用于分解不同分辨率子空间中的函数，其中缩放函数在构造\(L^{2}({\mathbb {R}})\)中的嵌套子空间中起着关键作用。本文在\(L^{2}({\mathbb {Q}}_p)\)中提出了相同的概括，称为帧多分辨率分析（FMRA）。因此FMRA是具有帧条件的多分辨率分析的推广。我们研究 FMRA 的各种属性，包括\(L^{2}({\mathbb {Q}}_p)\)中的特征。此外，还通过实例研究了帧缩放集。