Notes on harmonic analysis Part II: the Fourier Series

TL;DR

This paper introduces the fundamental concepts and theorems of Fourier Series within harmonic analysis, emphasizing clear explanations and complete proofs to aid students and researchers in understanding its mathematical foundations and applications.

Contribution

It provides comprehensive proofs and clear exposition of Fourier Series theory, making advanced harmonic analysis accessible to students and researchers.

Findings

Complete proofs of Fourier Series theorems

Clear explanations suitable for students and beginners

Emphasis on mathematical rigor and understanding

Abstract

Fourier Series is the second 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 approach 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.