L1空间上角度傅里叶变换性质的研究

傅立叶变换的研究是调和分析的基本问题，角度傅里叶变换是傅里叶变换的广义形式之一，在信号处理、密码学、光学等学科均有广泛应用，但角度傅里叶变换的性质在L1空间上鲜有系统研究。本课题旨在研究角度傅里叶变换在L1空间上的性质。具体包括：拟建立L1空间上角度傅里叶变换关于角度的半群性质和强连续性质；拟建立L1空间上角度傅里叶变换分别在时间和频率空间上与求导算子和乘法算子的复合变换的性质；拟建立L1空间上与角度傅里叶变换相关的卷积的性质。这些性质不仅揭示角度傅里叶变换对角度旋转的内在本质，也是经典傅里叶变换性质的推广，丰富广义时频分析理论。

The study of Fourier transform is the basic problem of harmonic analysis. The angular Fourier transform is one of the generalized forms of Fourier transform. It is widely used in signal processing, cryptography, optics and other disciplines. However, the properties of the angular Fourier transform are rarely systematically studied on L1 spaces. The purpose of this project is to study the properties of the angular Fourier transform on L1 spaces. It includes: to establish the semi-group property and strong continuous property of the angle Fourier transform on L1 spaces; In this paper, the properties of the compound transformation of the angular Fourier transform with the derivative operator and the multiplication operator in the time and frequency space will be established; The properties of convolution associated with the angular Fourier transform on L1 spaces will be established. These properties not only reveal the inherent nature of the angular Fourier transform on the angular rotation, but also extend the properties of the classical Fourier transform and enrich the theory of generalized time-frequency analysis.