连续扭曲时频表示——Coorbit空间与离散化
Continuous warped time-frequency representations - Coorbit spaces and discretization
AI总结：
本文提出一族适应多种非线性频率尺度的连续线性时频变换，通过扭曲函数构造原子，证明其可生成广义coorbit空间，并给出子采样系统构成原子分解和Banach框架的充分条件，同时涵盖经典时频表示。
摘要：
我们提出了一族新型的连续、线性时频变换，这些变换可适应多种（非线性）频率尺度。与经典的时频或时标表示类似，表示系数是通过与一族连续索引的时频原子进行内积获得的。这些原子由单个原型函数通过调制、平移和扭曲得到。这里所说的扭曲，是指根据一个双射的递增函数（即扭曲函数）进行非线性求值的过程。除了证明所得的积分变换满足某些基本但必要的性质（如连续性和可逆性）之外，我们还将证明，一大类扭曲函数可以产生广义的coorbit空间族，即其表示具有某种局部性的函数的Banach空间。此外，我们还获得了子采样扭曲时频系统构成原子分解和Banach框架的充分条件。为此，我们通过一个简单但关键的修改，将Fornasier和Rauhut先前提出的结果推广到更大类的函数系统。所提出的方法具有很大的灵活性，但通过选择特定的扭曲函数$Φ$，我们也能恢复经典的时频表示，例如$Φ(t) = ct$给出短时傅里叶变换，$Φ(t) = log_a(t)$给出小波变换。文中通过若干示例对此进行了说明。
英文摘要：
We present a novel family of continuous, linear time-frequency transforms adaptable to a multitude of (nonlinear) frequency scales. Similar to classical time-frequency or time-scale representations, the representation coefficients are obtained as inner products with the elements of a continuously indexed family of time-frequency atoms. These atoms are obtained from a single prototype function, by means of modulation, translation and warping. By warping we refer to the process of nonlinear evaluation according to a bijective, increasing function, the warping function. Besides showing that the resulting integral transforms fulfill certain basic, but essential properties, such as continuity and invertibility, we will show that a large subclass of warping functions gives rise to families of generalized coorbit spaces, i.e. Banach spaces of functions whose representations possess a certain localization. Furthermore, we obtain sufficient conditions for subsampled warped time-frequency systems to form atomic decompositions and Banach frames. To this end, we extend results previously presented by Fornasier and Rauhut to a larger class of function systems via a simple, but crucial modification. The proposed method allows for great flexibility, but by choosing particular warping functions $Φ$ we also recover classical time-frequency representations, e.g. $Φ(t) = ct$ provides the short-time Fourier transform and $Φ(t) = log_a(t)$ provides wavelet transforms. This is illustrated by a number of examples provided in the manuscript.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
