Linear and Non-linear Completeness

受过一些数学训练的人会熟悉标准三角函数。这些函数的形状和周期由圆的完全对称性决定。根据著名的傅立叶定理，任何合理形状的一维信号都可以按正弦和余弦分解成基频。这个项目的目的是为这些函数的扰动形成这个定理的不同版本。为了定义扰动，圆的对称性以某种受控的方式被打破，由一个典型的非线性微分方程表示。因此，通常的几何图形被替换为不同的几何图形，其属性由一组参数驱动。三角函数对应于方程的分岔点，当这些参数接近某一极限时得到。在p-拉普拉斯方程和非线性薛定谔方程的背景下，最近发现有可能建立解的精确版本的傅里叶定理。在某些情况下，这与关于正则函数的空间(Sobolevspace)如何嵌入到非正则函数的空间(Lebesguesspaces)的深入结果有关。关于信号如何分解成这些扰动模式的许多有趣的问题仍然悬而未决。其中一些问题的解决有望在非平稳数据的信号处理领域产生重大影响，因此该项目具有很大的创新潜力。具体目标将是提供一个坚实的数学框架，以将傅立叶定理的表述扩展到非线性环境，并推导出分叉点附近的基本微分方程的结果。该项目将开发变革性的研究工具，使一系列函数的线性完备性框架能够扩展到非线性环境。它将集中于与非线性特征值问题相关的给定的特征函数样本在潜在的Banach空间中是否在某种意义上是完备的问题。最近曝光了两个可能取得进展的领域。这些方程是Sobolevem层理理论中出现的非线性拉普拉斯本征值方程和分叉理论中所考虑的半线性谱方程。除了这两个具体案例，这一主题的研究图景似乎在很大程度上是未被探索的。

People with some mathematical training will be familiar with the standard trigonometric functions. The shape and periodicity of these functions are determined by the complete symmetry of the circle. According to a celebrated theorem of Fourier, one-dimensional signals of any reasonable shape can be decomposed into fundamental frequencies in terms of sine and cosine. The aim of this project is to formulate versions of this theorem for perturbations of these functions. To define the perturbations, the symmetries of the circle are broken in a certain controlled manner, represented by a canonical non-linear differential equation. Theusual geometry is therefore replaced with a different geometry, whose properties are driven by a set of parameters. The trigonometric functions correspond to a bifurcation point of the equation, obtained as these parameters approach a certain limit. In the context of the p-Laplacian and the non-linear Schroedinger equations, it has recently been discovered that it is possible to establish sharp versions of Fourier's Theorem for the solutions. In some instances, this is related to deep results about how spaces of regular functions (Sobolevspaces) are embedded into spaces of functions of less regularity (Lebesguespaces). Many interesting questions about how signals decompose into these perturbed modes remain open. Answers to some of these questions are expected to have significant impact in the field of signal processing of non-smooth data, so the project has substantial potential for innovation. The concrete goal will be to provide a solid mathematical framework to extend the formulation of Fourier's Theoremin to the non-linear setting and derive consequences for the underlying differential equations near the bifurcation points. This project will develop transformative research tools which will enable to extend the framework of linear completeness of a family of functions to the non-linear setting. It will focus on the question of whether a given sample of eigen functions, associated to a non-linear eigen value problem, is complete in a certain sense in the underlying Banach space. Two areas where progress may be possible have come to light recently. These are non-linear Laplacian eigen value equations arising in the theory of Sobolevem beddings and semi-linear spectral equations of the kind contemplated in bifurcation theory. Except from these two specific cases, the research landscape of the subject appears to be largely unexplored.