Structure of eigenvalues and eigenfunctions

傅里叶展开是分析偏微分方程的关键工具。不规则函数仍然可以展开为傅里叶级数，然后通过截断傅里叶级数得到收敛于原始不规则函数的光滑函数序列。这个想法——以及相关的技术——构成了该领域许多证明的关键部分。然而，截断的傅立叶级数——更一般地说，截断的特征函数展开式——在p不等于2的情况下可能不会收敛于Lp，这在很多情况下都是有问题的。基于仔细选择的傅立叶级数截断在p不等于2时收敛于Lp的观察，本项目将研究在一般域的特征函数展开（傅立叶级数的情况对应于正方形域的拉普拉斯）中，这可以在多大程度上进行。对于某些类型的三角域，我们已经得到了新的收敛性结果，并且正在开发一种分析在可控变形的情况下收敛性的技术。该研究属于数学分析领域，在连续介质力学、流体动力学和空气动力学等理论性方面具有潜在的应用前景。

Fourier expansions are a key tool in the analysis of partial differential equations. Irregular functions can still be expanded as Fourier series, and then by truncating the Fourier series one obtains a sequence of smooth functions that converges to the original irregular function. This idea - and related techniques - forms a key part of many proofs in the area. However, truncated Fourier series - and, more generally, truncated eigenfunction expansions - may not converge in Lp for p not equal to 2, and this can prove problematic in many situations. Based on the observation that carefully chosen truncations of Fourier series do converge in Lp when p is not 2, this project will investigate how much this can be carried over the eigenfunction expansions in general domains (the Fourier series case corresponding to the Laplacian in a square domain). We have obtained new convergence results for certain types of triangular domains and are developing a technique for analysing convergence as we deform the domain in controlled ways. The research lies in the area of Mathematical Analysis, with potential applications in the more theoretical aspects of Continuum Mechanics and Fluid dynamics and aerodynamics.