基于高维分数阶傅里叶变换反演的函数空间实变理论及其应用

函数空间实变理论、算子有界性质以及逼近性质属于调和分析研究的核心范畴. 项目拟在申请人与合作者关于一维分数阶傅里叶变换(简记为FRFT)可以分解为经典傅里叶变换和Chirp函数在时域和频域上复合运算的基础上，通过揭示FRFT在L1、L2函数空间中的有界性和逼近性质的差异，新定义一类分数阶Abel平均、分数阶Gauss平均作为工具，探讨高维FRFT的反演在函数空间中的逼近性质及其在信号恢复中的应用；并通过采样型离散FRFT进行数值计算和仿真实现；借助高维FRFT定义一类分数阶乘子，研究乘子定理和Littlewood-Paley定理；通过新乘子定理引入分数阶Riesz变换，定义一类分数阶Hardy空间，建立新的分数阶Sobolev、Besov、Triebel-Lizorkin空间实变理论，并将进一步应用于FRFT的反演问题研究中，这些研究的思想和方法将充实和丰富应用调和分析理论.

The real-variable theory of function spaces, the boundedness and approximation of operators have always been one of the core subjects of harmonic analysis. Based on the fact that one-dimensional fractional Fourier transform (abbreviated as FRFT) can be decomposed into the Fourier transform and the Chirp function in the time and frequency domains, this project intends to reveal the difference between the boundedness and approximation properties of FRFT in the L1 and L2 function spaces. With the help of a new class of fractional Abel averages and fractional Gauss average, we will discuss the approximation properties of high-dimensional FRFT inversion in Lebesgue and its application in signal recovery; and operate the numerical calculation and simulation through sampling DFRFT. We also use high-dimensional FRFT to define a class of fractional multiplier and establish the Littlewood-Paley theorem. We will introduce the fractional-order Riesz transform through the new multiplier theorem, define a class of fractional-order Hardy, Besov and Trieble-Lizorkin spaces and study the form and real-variable theory and further give applications in FRFT inversion. The ideas and methods of these studies will enrich the classics applied harmonic analysis theory.