de Gosson Maurice A
University of Vienna, Faculty of Mathematics, NuHAG, Austria.
Appl Comput Harmon Anal. 2015 Mar;38(2):196-221. doi: 10.1016/j.acha.2014.03.010.
Gabor frames can advantageously be redefined using the Heisenberg-Weyl operators familiar from harmonic analysis and quantum mechanics. Not only does this redefinition allow us to recover in a very simple way known results of symplectic covariance, but it immediately leads to the consideration of a general deformation scheme by Hamiltonian isotopies ( arbitrary paths of non-linear symplectic mappings passing through the identity). We will study in some detail an associated weak notion of Hamiltonian deformation of Gabor frames, using ideas from semiclassical physics involving coherent states and Gaussian approximations. We will thereafter discuss possible applications and extensions of our method, which can be viewed - as the title suggests - as the very first steps towards a general deformation theory for Gabor frames.
利用调和分析和量子力学中熟知的海森堡 - 外尔算子，可以对伽博框架进行有益的重新定义。这种重新定义不仅使我们能够以非常简单的方式恢复辛协变性的已知结果，而且立即引发了通过哈密顿同伦（经过恒等映射的非线性辛映射的任意路径）来考虑一种一般的变形方案。我们将使用涉及相干态和高斯近似的半经典物理思想，详细研究伽博框架的哈密顿变形的相关弱概念。此后，我们将讨论我们方法的可能应用和扩展，正如标题所暗示的，这可以被视为迈向伽博框架一般变形理论的第一步。
