ABSTRACT
This chapter introduces the concept of Fourier transform which is part and parcel of the theoretical study of partial differential equations. The purpose is to show some applications of some of the basic theorems in the subject of measure and integration, such as dominated convergence theorem and Fubini’s theorem, to another branch of analysis. It discusses the case of Fourier transform on L1(R) where some basic properties are highlighted and explains the condition of Fourier transform as a linear operator. One question one may ask is whether f can be recovered from f^. The answer is affirmative if f^ also belongs to L1(R). The chapter shows how the Fourier inversion theorem is proved, and discusses Fourier-Plancherel transform and shows that it is surjective as well besides being a linear isometry.
