Structure of eigenvalues and eigenfunctions

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 L^p 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 L^p 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). The results will be interesting in their own right, and should produce techniques useful in the analysis of a wide class of PDE models. The research lies in the area of Mathematical Analysis, with potential applications in the more theoretical aspects of Continuum Mechanics and Fluid dyanmics & aerodynamics.