Wigner analysis of operators. Part I: Pseudodifferential operators and wave fronts
Wigner analysis of operators. Part I: Pseudodifferential operators and wave fronts
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DOI：
10.1016/j.acha.2022.01.003
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发表时间：
2022-02-17
影响因子：
2.5
通讯作者：
Rodino, Luigi
中科院分区：
文献类型：
--
作者：
Cordero, Elena;Rodino, Luigi
我们对线性算子进行Wigner分析。也就是说，标准时频表示的短时傅立叶变换(STFT)被定义为W-A(F)=Mu(A)(f循环次数f)的A-Wigner分布所取代，其中A是4d×4d辛矩阵，Mu(A)是相关联的亚辛算子。基本的例子是所谓的陶-维格纳分布。当tau是(0，1)的元素时，这种表示为调制空间提供了新的特征。此外，它们还可以有效地用于研究符号在Sjostrand类(特别是在Hormander类S-0，0(0))中的伪微分算子的非对角衰减性。这种新颖性依赖于通过亚普勒运算符定义时间-频率表示，开发概念框架，并为对量化过程的新理解铺平道路。利用Wigner波前集，我们推导了拟微分算子的微局域性质。最后，我们将Wigner与全球霍曼德波前集进行了比较，并确定了Wigner波前中可能存在鬼影区。在论文的第二部分将给出傅里叶积分算子和薛定谔方程的应用(C)2022 Elsevier Inc.保留所有权利。
We perform Wigner analysis of linear operators. Namely, the standard time frequency representation Short-time Fourier Transform (STFT) is replaced by the A-Wigner distribution defined by W-A(f) = mu(A)(f circle times f), where A is a 4d x 4d symplectic matrix and mu(A) is an associate metaplectic operator. Basic examples are given by the so-called tau-Wigner distributions. Such representations provide a new characterization for modulation spaces when tau is an element of (0, 1). Furthermore, they can be efficiently employed in the study of the off-diagonal decay for pseudodifferential operators with symbols in the Sjostrand class (in particular, in the Hormander class S-0,0(0)). The novelty relies on defining time-frequency representations via metaplectic operators, developing a conceptual framework and paving the way for a new understanding of quantization procedures. We deduce micro-local properties for pseudodifferential operators in terms of the Wigner wave front set. Finally, we compare the Wigner with the global Hormander wave front set and identify the possible presence of a ghost region in the Wigner wave front. In the second part of the paper applications to Fourier integral operators and Schrodinger equations will be given (c) 2022 Elsevier Inc. All rights reserved.