径向调制空间的径向时频分析与嵌入
Radial Time-Frequency Analysis and Embeddings of Radial Modulation Spaces
摘要：
本文对径向$L^2$ -函数空间$L^2_{rad}(\R^d)$，更一般地说，对由径向分布组成的调制空间的子空间，构造了Gabor型框架。因此，每个框架元件本身是径向函数。这种构造是基于所谓的Feichtinger-Gröchenig理论的推广-有时也称为共轨空间理论-这是在早期的文章中发展起来的。我们证明了这种新的Gabor框架在线性和非线性逼近方面比通常的Gabor框架在径向函数逼近方面有更好的表现。此外，我们得到新的嵌入定理限制不变向量（函数）的共轨空间，并将其应用到调制空间的径向分布。作为一个特殊情况，这个结果意味着限定为径向函数的Feichtinger代数$(S_0)_{rad}(\R^d) = M^1_{rad}(\R^d)$嵌入到索伯列夫空间$H^{(d-1)/2}_{rad}(\R^d)$中。此外，对于$d\geq 2$，嵌入$(S_0)_{rad}(\R^d) \hookrightarrow L^2_{rad}(\R^d)$是紧凑的。
英文摘要：
In this paper we construct frames of Gabor type for the space $L^2_{rad}(\R^d)$ of radial $L^2$-functions, and more generally, for subspaces of modulation spaces consisting of radial distributions. Hereby, each frame element itself is a radial function. This construction is based on a generalization of the so called Feichtinger-Gröchenig theory -- sometimes also called coorbit space theory -- which was developed in an earlier article. We show that this new type of Gabor frames behaves better in linear and non-linear approximation in a certain sense than usual Gabor frames when approximating a radial function. Moreover, we derive new embedding theorems for coorbit spaces restricted to invariant vectors (functions) and apply them to modulation spaces of radial distributions. As a special case this result implies that the Feichtinger algebra $(S_0)_{rad}(\R^d) = M^1_{rad}(\R^d)$ restricted to radial functions is embedded into the Sobolev space $H^{(d-1)/2}_{rad}(\R^d)$. Moreover, for $d\geq 2$ the embedding $(S_0)_{rad}(\R^d) \hookrightarrow L^2_{rad}(\R^d)$ is compact.
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