Beurling Dimension of Gabor Pseudoframes for Affine Subspaces
Beurling Dimension of Gabor Pseudoframes for Affine Subspaces
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DOI：
10.1007/s00041-008-9026-0
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发表时间：
2008-04
影响因子：
1.2
通讯作者：
W. Czaja;Gitta Kutyniok;D. Speegle
中科院分区：
文献类型：
--
作者：
W. Czaja;Gitta Kutyniok;D. Speegle
最近，Li和Ogawa引入了子空间伪框架，作为分析低维数据的工具，具有分析序列和对偶序列的任意灵活性。本文通过关注其相关参数集的几何性质来研究仿射子空间的Gabor伪框架。本文首先引入了一个新的概念的离散子集的Beurling维数通过使用一定的广义Beurling密度。我们提出了几个性质的Beurling尺寸包括与其他概念的尺寸显示比较，例如，我们的概念包括质量尺寸作为一个特例。然后证明了仿射子空间的Gabor伪框架满足一定的齐次逼近性质，这意味着伪框架中的元素的逼近在时频移动下不变性.本文的主要结果是利用仿射子空间的Gabor伪框架的参数集的Beurling维数对它们进行分类.这为我们提供了，特别是，与奈奎斯特维数分离的伪帧的参数集从那些非伪帧，并链接到一个固定值的伪Riesz序列的参数集。这些结果甚至是新的特殊情况下的仿射子空间的Gabor框架。
Pseudoframes for subspaces have been recently introduced by Li and Ogawa as a tool to analyze lower dimensional data with arbitrary flexibility of both the analyzing and the dual sequence.In this paper we study Gabor pseudoframes for affine subspaces by focusing on geometrical properties of their associated sets of parameters. We first introduce a new notion of Beurling dimension for discrete subsets of ℝdby employing a certain generalized Beurling density. We present several properties of Beurling dimension including a comparison with other notions of dimension showing, for instance, that our notion includes the mass dimension as a special case. Then we prove that Gabor pseudoframes for affine subspaces satisfy a certain Homogeneous Approximation Property, which implies invariance under time–frequency shifts of an approximation by elements from the pseudoframe.The main result of this paper is a classification of Gabor pseudoframes for affine subspaces by means of the Beurling dimension of their sets of parameters. This provides us, in particular, with a Nyquist dimension which separates sets of parameters of pseudoframes from those of non-pseudoframes and which links a fixed value to sets of parameters of pseudo-Riesz sequences. These results are even new for the special case of Gabor frames for an affine subspace.