Linear and Non-linear Completeness

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.