Regularity for complete and minimal Gabor systems on a lattice
Regularity for complete and minimal Gabor systems on a lattice
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格上完整和最小 Gabor 系统的正则性
DOI：
10.1215/ijm/1290435340
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发表时间：
2009
影响因子：
0.6
通讯作者：
A. Powell
中科院分区：
文献类型：
--
作者：
C. Heil;A. Powell
在L(R)中证明了完备极小的Gabor系统G(g，1，1)的巴利安-洛定理的非对称加权推广.对于g∈L(R)，证明了如果3<；p≤4≤q<；∞满足3/p+1/q=1，且R|x||g(X)|dx<；∞和R|ξ||bg(ξ)|2 dξ<；∞，则G(g，1，1)={Eg(x−k)}k，n∈Z在L(R)中不是完全极小的.对于端点情形(p，q)=(3，∞)，证明了如果g∈L(R)是紧支撑的，且R|ξ||bg(ξ)|2 dξ<；∞，则G(g，1，1)在L(R)中是不完备且极小的.这些定理推广了Daubechies和Janssen在(p，q)=(4，4)情形下的工作。文中还给出了进一步的改进和优化实例。
Nonsymmetrically weighted extensions of the Balian–Low theorem are proved for Gabor systems G(g, 1, 1) that are complete and minimal in L(R). For g ∈ L(R), it is proven that if 3 < p ≤ 4 ≤ q < ∞ satisfy 3/p + 1/q = 1 and R |x| |g(x)| dx < ∞ and R |ξ| |bg(ξ)|2 dξ < ∞ then G(g, 1, 1) = {eg(x − k)}k,n∈Z cannot be complete and minimal in L(R). For the endpoint case (p, q) = (3,∞), it is proved that if g ∈ L(R) is compactly supported and R |ξ| |bg(ξ)|2 dξ < ∞ then G(g, 1, 1) is not complete and minimal in L(R). These theorems extend the work of Daubechies and Janssen from the case (p, q) = (4, 4). Further refinements and optimal examples are also provided.