The Malgrange-Ehrenpreis theorem in distribution theory

This thesis explores the Malgrange-Ehrenpreis Theorem in the theory of distribution via an expository approach based on the text Introduction to the Theory of Distribution by G. Friedlander and M. Joshi. I will be focusing on the understanding of the proof of the Malgrange-Ehrenpries Theorem. In many places, I have closely followed the presentation of Friedlander's and Joshi's monograph. We begin with the definition and the properties of distributions. Next, we study Fourier transforms and its properties. We look at the Schwartz space and consider its dual space. We conclude the section by investigating the Fourier transform of several notable distributions, including the Dirac-delta distribution and the signum distribution. Finally, we introduce Fourier-Laplace transforms and examine the Paley-Wiener-Schwartz estimate. This estimate describes the polynomial decay property of the Fourier-Laplace trans- form of test functions. We conclude with a special case of Malgrange-Ehrenpreis Theorem and analyze its complex proof.