Matrix Representation of Magnetic Pseudo-Differential Operators via Tight Gabor Frames

IF 1.2 3区数学 Q2 MATHEMATICS, APPLIED

Journal of Fourier Analysis and Applications Pub Date : 2024-04-04 DOI:10.1007/s00041-024-10072-4

Horia D. Cornean, Bernard Helffer, Radu Purice

引用次数: 0

Abstract

In this paper we use some ideas from [12, 13] and consider the description of Hörmander type pseudo-differential operators on \(\mathbb {R}^d\) (\(d\ge 1\)), including the case of the magnetic pseudo-differential operators introduced in [15, 16], with respect to a tight Gabor frame. We show that all these operators can be identified with some infinitely dimensional matrices whose elements are strongly localized near the diagonal. Using this matrix representation, one can give short and elegant proofs to classical results like the Calderón-Vaillancourt theorem and Beals’ commutator criterion, and also establish local trace-class criteria.

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通过紧密 Gabor 框架对磁伪微分算子进行矩阵表示

在本文中，我们使用了 [12, 13] 中的一些观点，并考虑了关于紧 Gabor 框架的 Hörmander 型伪微分算子在 \(\mathbb {R}^d\) (\(d\ge 1\)) 上的描述，包括 [15, 16] 中引入的磁性伪微分算子的情况。我们证明，所有这些算子都可以与一些无限维矩阵相鉴别，这些矩阵的元素在对角线附近强局部化。利用这种矩阵表示，我们可以对卡尔德龙-瓦扬库尔定理和比尔斯换元准则等经典结果给出简短而优雅的证明，还可以建立局部迹类准则。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Fourier Analysis and Applications 数学-应用数学

CiteScore

2.10

自引率

16.70%

发文量

审稿时长

6-12 weeks

期刊介绍： The Journal of Fourier Analysis and Applications will publish results in Fourier analysis, as well as applicable mathematics having a significant Fourier analytic component. Appropriate manuscripts at the highest research level will be accepted for publication. Because of the extensive, intricate, and fundamental relationship between Fourier analysis and so many other subjects, selected and readable surveys will also be published. These surveys will include historical articles, research tutorials, and expositions of specific topics. TheJournal of Fourier Analysis and Applications will provide a perspective and means for centralizing and disseminating new information from the vantage point of Fourier analysis. The breadth of Fourier analysis and diversity of its applicability require that each paper should contain a clear and motivated introduction, which is accessible to all of our readers. Areas of applications include the following: antenna theory * crystallography * fast algorithms * Gabor theory and applications * image processing * number theory * optics * partial differential equations * prediction theory * radar applications * sampling theory * spectral estimation * speech processing * stochastic processes * time-frequency analysis * time series * tomography * turbulence * uncertainty principles * wavelet theory and applications