Notes on Harmonic Analysis Part I: The Fourier Transform

TL;DR

This monograph introduces the Fourier Transform as a fundamental tool in harmonic analysis, emphasizing complete proofs and clarity to aid students and researchers in understanding its core concepts and applications.

Contribution

It provides a detailed, proof-based introduction to Fourier Transforms within harmonic analysis, aimed at students with Lebesgue measure and integral background.

Findings

Complete proofs of Fourier Transform theorems

Clear explanations suitable for students and beginners

Foundational understanding for applications in sciences and engineering

Abstract

Fourier Transforms is a first in a series of monographs we present on harmonic analysis. Harmonic analysis is one of the most fascinating areas of research in mathematics. Its centrality in the development of many areas of mathematics such as partial differential equations and integration theory and its many and diverse applications in sciences and engineering fields makes it an attractive field of study and research. The purpose of these notes is to introduce the basic ideas and theorems of the subject to students of mathematics, physics or engineering sciences. Our goal is to illustrate the topics with utmost clarity and accuracy, readily understandable by the students or interested readers. Rather than providing just the outlines or sketches of the proofs, we have actually provided the complete proofs of all theorems. This will illuminate the necessary steps taken and the machinery used to complete each proof. The prerequisite for understanding the topics presented is the knowledge of Lebesgue measure and integral. This will provide ample mathematical background for an advanced undergraduate or a graduate student in mathematics.