维纳引理及其相关问题的研究

如果函数f(x)有绝对收敛的富里埃级数，且在实数域内处处不为零，那么1/f(x)也有绝对收敛的富里埃级数。这就是经典的维纳引理。维纳引理阐明了无穷矩阵和积分算子的局部化特性在反演下是保留的。经典的维纳引理及其各种推广形式是非常重要的，它们在数值分析、稀疏优化、小波理论、框架理论、采样理论以及工程技术等方面有着大量的应用。.本项目主要目标是研究无穷矩阵和积分算子的维纳引理；并进一步地探讨维纳引理更多的形式；建立齐次空间上Bessel位势算子的维纳引理；建立富里埃分析中几种常见的积分算子的谱不变性和稳定性；研究指标集为紧流形或者具有群结构时，无穷矩阵类的逆闭性和稳定性，并进一步探讨它们在采样理论、稀疏优化等发面的应用。.本项目将丰富维纳引理的研究内容，推动其理论和应用的发展。

If a periodic function f(x) has an absolutely convergent Fourier series and never vanishes, then 1/f(x) has an absolutely convergent Fourier series. It is called the classical Wiener’s lemma. Wiener's lemma states that the localization of infinite matrices and integral operators are preserved under inversion. .The classical Wiener’s lemma and its various generalizations are important，and they have numerous applications in numerical analysis, wavelet theory, frame theory, sampling theory and in engineering science including signal processing and communication engineering. .The primary objective of this project is to study the Wiener’s lemma for infinite matrices and integral operators. Furthermore, we want to study more form of Wiener’s lemma；Establishing the Wiener's lemma for Bessel potential operators on homogeneous spaces; Establishing Wiener's lemma for some kind of integral operators in Fourier Analysis; Discussing the closedness and stability of infinite matrices when their index set is the compact manifolds or has a group structure. And further to explore their application in sampling theory, and the sparse optimization etc..This project will enrich the research contents of Wiener's lemma, and promote the development of its theory and application.