Analysis I: Fourier Series and Partial Differential Equations

Description of Course Goals and Curriculum

Fourier Analysis is a class about the theory of Fourier Analysis and its application to other mathematical concepts. The class will go through all of the steps that lead to the establishment of this theory and prove them rigorously. There will be four big modules: Fourier sums, which is Fourier analysis of 2-periodic functions; Fourier transform, which is Fourier analysis of decaying functions on the real line; Fourier transform of functions on d-dimensions; Fourier analysis of functions on finite groups. Two main theorems will be recurring on each of these modules and most of the module will be concerned proving these two theorems. For each module there will also be applications, such as using Fourier analysis to solve PDE-s and the proof of the Dirichlet theorem, which says that "in every progression of the from qk + b, k = 1, 2, 3.., q and b coprime, there are infinitely many prime numbers". Dirichlet's theorem takes a chapter to prove and is a very interesting and powerful example of the combination of number theory and Fourier Analysis. Besides equipping the students with an in-depth knowledge of the theory of Fourier Analysis, in this class you learn about some very important pde-s such as the wave equation and the heat equation and develop skills that will help you in other analysis classes.

Learning From Classroom Instruction

Prof. Ionescu gives very detailed proofs in his lectures. He never "talks over" parts of the proof, but writes everything on the board. This type of lecturing makes it possible to take very detailed notes and to understand almost all of the material in class or after reading the notes.  During class he asks for student participation, mostly by asking questions about proofs we have done before. Therefore a good practice is to read some of your previous notes before lecture, as it will help both the memorization of the material and class participation.  

Learning For and From Assignments

This class has six problem sets, a midterm and a final exam.

1. Problem Sets

Prof. Ionescu only assigned 6 problem sets but they were longer than problem sets for other math classes and we had about 1 week and a half to complete each. It would be very subjective to say how hard students find these problem sets, but they do take longer than problem sets for any other math class. Working in groups can reduce the time spent on each problem set. Prof. Ionescu is also very helpful in office hours. Although long, the problems assigned are very helpful to the understanding of the material and provide good practice in problem-solving. In some of them you even develop a small theory of its own.

2. Midterm

Prof. Ionescu described the midterm as easier than the problem sets and that was indeed the case. Differently from most math classes the midterm was more of a measure of how well you knew the theory and required less creativity in solving the problems than the psets. One cheat-sheet was allowed however many students did not rely on it that much: the midterm has a short time so knowing the theory on the top of your head is important. To study for the midterm, besides understanding all of the theory one needs to be able to reproduce on the top of their head the main theorems. There might not be time to practice with exercises for the midterm however because of how Prof. Ionescu wrote the exam knowing the theory would be enough.

3. Final

Differently from the midterm, the final is about the same level of hardship as the problem sets, so it requires more creativity when solving problems.  It also has about two or three proofs from the textbook. Firstly, the proofs from the book were the proofs of very important theorems so there is no need to worry about learning each single proof. That said, it is good to know the two or three most important steps of each proof since you could use those moves when solving other problems. It is especially useful to spend time practicing problems from the book that were not assigned on the problem sets, since most of the points on the exam were for solving problems that we had not seen before.

External Resources

The textbook, written by Professor Stein, provides everything that you need to know about the class and more.

What Students Should Know About This Course For Purposes Of Course Selection

It should be noted that this class will not include "real world" applications of Fourier Analysis, but only applications to other mathematical concepts. There is an ELE class that teaches the more tangible applications, if that is what you are looking for. This class can be used as a departmental to fulfill the Real Analysis requirement for math students.