Fourier transform and some of its applications

Abstract

This research is primarily based on Alazard’s book, which serves as the main source for the definitions, properties, and theorems related to Fourier transforms. We thoroughly explore and provide proofs and explanations of various concepts presented in Alazard’s work. While Alazard’s book forms the foundation of our study, we also seek to enrich our under- standing of Fourier transforms and their applications by consulting other relevant sources. These sources, including works by L. Grafakos and Y. Tsutsumi, provide valuable insights that complement and enhance the analyses presented in our research. By combining information from multiple sources, we aim to present a comprehensive and well-rounded exploration of Fourier transforms, ensuring that our research is both rigorous and informed.