Ebba Nauwelaerts de Agé: An introduction to the Fourier transform in the Schwartz space and some applications to PDEs

Abstract.

This thesis is an introduction to the Fourier transform on functions in the Schwartz space. The definition of a function being in the Schwartz space will be stated and some properties these type of functions have, e.g. the behavior of the convolution of two functions, will be explained. Thereafter, the Fourier transform is introduced and also some important theorems, like the Fourier Inversion Formula. Finally, we will apply this theory on two famous partial differential equations, namely the wave equation and the heat equation. The wave equation will be solved in a compact form in one dimension, and the heat equation will be solved in n dimensions.