论文标题
带有HAAR窗口的Gabor系统的框架集
Frame set for Gabor systems with Haar window
论文作者
论文摘要
我们显示了Gabor系统$ \ MATHCAL {g;α，β）的框架集的完整结构：= \ {e^{ - 2πimβ\ cdot} g（\ cdot-nα）：m，n \ in \ bbb z \ z \} $ g =-χ_ {[ - 1/2,0）}+χ_ {[0,1/2）} $。本文的策略是在单位圆上介绍分段线性转换$ \ MATHCAL {M} $，并为其（对称）最大不变集提供结构的完整表征。这种转变与著名的斯坦豪斯的三个差距定理有关，这可能具有独立的兴趣。此外，改进了Gabor框架上的经典标准，这使我们能够为Gabor System $ \ Mathcal {g;α，β）建立{a}的必要条件，以成为框架，即，对称的对称的不可行的变换$ \ nathcal $ \ nathcal $ \ nathcal $ \ mathcal $} $是空的。 与以前的研究相比，本文提供了一个独立的环境，可以通过新的角度研究Gabor框架，其中包括这里开发的技术是新的，读者可以彻底理解所有证据，而无需参考先前文献中已知的结果。
We show the full structure of the frame set for the Gabor system $\mathcal{G}(g;α,β):=\{e^{-2πi mβ\cdot}g(\cdot-nα):m,n\in\Bbb Z\}$ with the window being the Haar function $g=-χ_{[-1/2,0)}+χ_{[0,1/2)}$. The strategy of this paper is to introduce the piecewise linear transformation $\mathcal{M}$ on the unit circle, and to provide a complete characterization of structures for its (symmetric) maximal invariant sets. This transformation is related to the famous three gap theorem of Steinhaus which may be of independent interest. Furthermore, a classical criterion on Gabor frames is improved, which allows us to establish {a} necessary and sufficient condition for the Gabor system $\mathcal{G}(g;α,β)$ to be a frame, i.e., the symmetric invariant set of the transformation $\mathcal{M}$ is empty. Compared with the previous studies, the present paper provides a self-contained environment to study Gabor frames by a new perspective, which includes that the techniques developed here are new and all the proofs could be understood thoroughly by the readers without reference to the known results in the previous literature.
