The special atom space and Haar wavelets in higher dimensions
The special atom space and Haar wavelets in higher dimensions
复制标题
特殊原子空间和高维Haar小波
DOI：
10.1515/dema-2020-0011
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发表时间：
2020
影响因子：
2
通讯作者：
M. Ndiaye
中科院分区：
文献类型：
--
作者：
E. Kwessi;Geraldo de Souza;N. Djitté;M. Ndiaye
在这篇文章中，我们将重温由Geraldo De Souza和Richard O 'Neil在20世纪80年代初引入的特殊原子空间。在他们的入门作品和后来的补充作品中，空间主要是在真实的线上研究的。与Orlicz， Lipschitz， Lebesgue和Lorentz等空间的有趣性质和联系使这些空间在更高维度上的探索成熟。在本文中，我们将这个定义扩展到平面和空间，并展示了几乎所有有趣的性质，如它们的Banach结构、Hölder’s型不等式和对偶性都被保留了下来。特别地，特殊原子空间的对偶空间是函数的Lipschitz空间和广义Lipschitz空间在高维上的自然扩展。我们指出，这种扩展可以允许研究范围广泛的问题，包括一个连接，导致似乎是Haar函数的新定义，Haar小波，以及平面和空间上的小波。
Abstract In this note, we will revisit the special atom space introduced in the early 1980s by Geraldo De Souza and Richard O’Neil. In their introductory work and in later additions, the space was mostly studied on the real line. Interesting properties and connections to spaces such as Orlicz, Lipschitz, Lebesgue, and Lorentz spaces made these spaces ripe for exploration in higher dimensions. In this article, we extend this definition to the plane and space and show that almost all the interesting properties such as their Banach structure, Hölder’s-type inequalities, and duality are preserved. In particular, dual spaces of special atom spaces are natural extension of Lipschitz and generalized Lipschitz spaces of functions in higher dimensions. We make the point that this extension could allow for the study of a wide range of problems including a connection that leads to what seems to be a new definition of Haar functions, Haar wavelets, and wavelets on the plane and on the space.