Wavelet Coorbit Spaces viewed as Decomposition Spaces
Wavelet Coorbit Spaces viewed as Decomposition Spaces
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DOI：
10.1016/j.jfa.2015.03.019
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发表时间：
2014-04
期刊：
影响因子：
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通讯作者：
Hartmut Fuhr;F. Voigtlaender
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文献类型：
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作者：
Hartmut Fuhr;F. Voigtlaender
本文证明了Fourier变换诱导了混合加权Lebesgue空间Lvp，q的Feichtinger和Gröchenig定义的关于半直积Rd <$H的拟正则表示的共轨空间与某些分解空间D之间的同构（Q，L p，u q）（本质上是由Feichtinger和Gröbner引入的）其中函数的局部化“部分”在F L p-范数中测量。这种等价性在几个方面是有用的：它提供了对小波共轨空间的傅立叶分析理解，并且它允许在一个共同的框架中讨论与不同伸缩群相关的共轨空间。作为这些点的一个例子，我们包括一个简短的讨论膨胀不变性性质的coorbit空间与不同类型的膨胀组。
In this paper we show that the Fourier transform induces an isomorphism between the coorbit spaces defined by Feichtinger and Gröchenig of the mixed, weighted Lebesgue spaces L v p, q with respect to the quasi-regular representation of a semi-direct product R d⋊ H with suitably chosen dilation group H, and certain decomposition spaces D (Q, L p, ℓ u q)(essentially as introduced by Feichtinger and Gröbner) where the localized “parts” of a function are measured in the F L p-norm. This equivalence is useful in several ways: It provides access to a Fourier-analytic understanding of wavelet coorbit spaces, and it allows to discuss coorbit spaces associated to different dilation groups in a common framework. As an illustration of these points, we include a short discussion of dilation invariance properties of coorbit spaces associated to different types of dilation groups.