超Gabor框架中的矩阵格变换算子
Matrix Lattice Transform Operator for Super Gabor Frames
摘要: 在 空间，对于时频格 ，存在超Gabor框架的充要条件是 。在 空间，已知如何构造一个 的超Gabor框架。本文的目的是构造 意义下的超Gabor框架(特别是当A = I的时候，存在正交的超Gabor框架， 是满秩的格)，这样就能构造出任何满秩的格的超Gabor框架，并且给出矩阵格变换算子的概念和充要条件。在构建超Gabor框架的方法之上，矩阵格变换算子能帮助我们更好的理解超Gabor框架的基础理论。最后，给出了关于构造超Gabor框架的还有矩阵格变换算子的例子。
Abstract: For a time frequency lattice , it is known that a super Gabor frame of length L exists if and only if in . In , we know that we can construct a super Gabor frame for . We can construct a super Gabor frame (especially orthonormal super Gabor frame) for any full rank lattices by our result. And we find matrix lattice transform operator and it is necessary and sufficient condition. Matrix lattice transform operators can be used to obtain new super Gabor frames and can help us better understand the basic theory of super Gabor frames theorem based on our method of constructing a super Gabor frame. At last, we give some examples about how to construct super Gabor frames based on our method and the application of matrix lattice transform operator.
文章引用：肖炳环, 李忠艳. 超Gabor框架中的矩阵格变换算子[J]. 理论数学, 2018, 8(4): 398-406. https://doi.org/10.12677/PM.2018.84053

参考文献

[1] Regev, O. (2004) New Lattice-Based Cryptographic Constructions. Journal of the ACM, 51, 899-942. [Google Scholar] [CrossRef] 
[2] Balan, R. (1999) Density and Redundancy of the Noncoherent Weyl-Heisenberg Superframes. In: Contemporary Mathematics, 247, American Mathematical Society, Providence, RI, 29-41. [Google Scholar] [CrossRef] 
[3] Gabardo, J.P. and Han, D. (2002) Aspects of Garbor Analysis and Operator Al-gebras. In: Feichtinger, H.G. and Strohmer, T., Eds., Advances in Gabor Analysis, Birkhauser, Boston, 129-150.
[4] Han, D. and Wang, Y. (2001) Lattice Tiling and the Weyl—Heisenberg Frames. Geometric Functional Analysis GAFA, 11, 742-758. [Google Scholar] [CrossRef] 
[5] Li, Z.Y. and Han, D.G. (2010) Constructing Super Gabor Frames: The Rational Time-Frequency Lattice Case. Science China, 53, 3179-3186. [Google Scholar] [CrossRef] 
[6] Gu, Q. and Han, D. (2003) Functional Gabor Frame Multipliers. Journal of Geometric Analysis, 13, 467-478. [Google Scholar] [CrossRef] 
[7] Gabardo, J.P. and Han, D. (2003) Frame Representations for Group-Like Unitary Operator Systems. The Journal of Operator Theory, 49, 223-244.
[8] Shi, X.L. and Chen, F. (2007) Necessary Conditions for Ga-bor Frames. Science in China, 50, 276-284. [Google Scholar] [CrossRef] 
[9] Casazza, P.G. (2001) Modern Tools for Weyl-Heisenberg (Gabor) Frame Theory. Advances in Imaging. Electron Physics, 115, 1-127. [Google Scholar] [CrossRef] 

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