Tight frames generated by finite nonabelian groups
Tight frames generated by finite nonabelian groups
复制标题
DOI：
10.1007/s11075-008-9167-x
复制
发表时间：
2008-03
影响因子：
2.1
通讯作者：
Richard Vale;S. Waldron
中科院分区：
文献类型：
--
作者：
Richard Vale;S. Waldron
设n ≥d是有限维Hilbert空间，如有限信号空间或多元正交多项式空间。存在有限个n个向量的紧框架，对于这些紧框架，可以得到单个向量在（框架对称的）阿贝尔群G的酉作用下的轨道。这些所谓的调和标架或几何一致标架中的每一个都可以用一种简单的方法从G的特征标表中得到。这些框架用于信号处理和信息论。对于一个非交换群G，一般存在不可数的n个向量的不等价紧框架，它们可以作为这样的G-轨道得到。然而，通过添加一个额外的自然对称条件（自动持有，如果G是阿贝尔），我们得到了有限类这样的框架，可以从字符表G在类似的方式构造的调和框架。这是通过将每个G-轨道与群代数的一个元素（通过它的Gramian）标识，在群代数中施加条件，然后描述相应的紧框架类来完成的。
Letbe a Hilbert space of finite dimensiond, such as the finite signalsℓ2(d) or a space of multivariate orthogonal polynomials, andn≥d. There is a finite number of tight frames ofnvectors forwhich can be obtained as the orbit of a single vector under the unitary action of an abelian groupG(of symmetries of the frame). Each of these so calledharmonic framesorgeometrically uniform framescan be obtained from the character table ofGin a simple way. These frames are used in signal processing and information theory. For a nonabelian groupGthere are in general uncountably many inequivalent tight frames ofnvectors forwhich can be obtained as such aG-orbit. However, by adding an additional natural symmetry condition (which automatically holds ifGis abelian), we obtain a finite class of such frames which can be constructed from the character table ofGin a similar fashion to the harmonic frames. This is done by identifying eachG-orbit with an element of the group algebra ℂG(via its Gramian), imposing the condition in the group algebra, and then describing the corresponding class of tight frames.