Analysis IV - Lebesgue measure, Fourier analysis

Teacher: Prof. Maria Colombo

Assistants: Roberto Colombo, Giulia Mescolini

Student assitants: Patrick-Cristian Dan, Mathis Duguin, Francesco Pio Numero, Giacinta Vescovo

Lectures

The lectures will be taught at the blackboard. In case you miss a class, you may find helpful to find the similar material discussed in 2021 in the videos at the link https://mediaspace.epfl.ch/channel/MATH-205+Analysis+IV/30037.

Exercise Sessions

Every Monday at 10am there will be a new exercise sheet available on the Moodle page and the solutions of the exercise sheet of the previous week. The exercise session will start with some quiz questions on the lectures of the previous week and include a Q&A session on the exercise sheet of the week.

The format should be a PDF, and the file name should be: YourName_YourSurname_WeekX.pdf. For example, an exercise that is to be handed for week 1 will have the file name Jane_Doe_Week1.pdf

Course content  

• Chapter 1. Lebesgue measure
  
• Chapter 2. Lebesgue integration, Monotone and dominated convergence theorems
  
• Chapter 3. L^p spaces
  
• 
Chapter 4. Fourier analysis

  
• Chapter 5. Fourier transform

  
• Chapter 6. Applications to partial differential equations
  


For more details, you can refer to the course book.


Material and reference books
Students are expected to learn the material discussed in class. The lectures, in turn, will be based mainly on the book of Tao (Chapters 7, 8, 5) and on the Polycopié of Dacorogna (Chapters 16, 17, 18, 19):

• 

Polycopié of B. Dacorogna (see pdf below)
  
• T. Tao: "Analysis II" (available also online and at the library - also the electronic version)

The polycopié de cours in LaTeX, inspired by these two sources and the previous editions of the course, are available: for every chapter presented in the course, the corresponding section will be uploaded on Moodle shortly before the discussion of the material in class.

Stein’s books are also a precious source of interesting exercises and for deepening your understanding:

• E. Stein: "Real analysis: measure theory, integration, and Hilbert spaces"
  
• E. Stein: "Fourier analysis: an introduction"

S.D. Chatterji: "Cours d'analyse 1 et 3" PPUR
S.D. Chatterji: "Equations différentielles ordinaires et aux dérivées partielles"
B. Dacorogna: "Analyse avancée pour ingénieurs"

Exam
The exam lasts 3 hours. It contains a first exercise with several multiple choice questions, evaluated 0=not given or wrong, 1=correct. This first exercise is followed by other exercises and theoretical questions with open answers. The list of statements and proofs of the course that can be asked as theoretical questions at the exam will be uploaded on this Moodle page.

You are allowed to bring to the exam a one-sided, A4 paper with notes handwritten by you personally and "physically". In particular, it is not allowed to bring printed notes from another source (e.g. iPad, graphic tablet etc). 

Midterm
The midterm lasts 1.5 hours and its format is half of the final exam (MCQ, exercises, theoretical questions). Correspondingly, you can bring a one-sided, A5 (namely, half of an A4) paper with notes handwritten by you personally and "physically". 

The final grade will be determined as follows: max{ Final grade , 0.4 Midterm + 0.6 Final grade}

The midterm will take place on April 9th, at 13.15.

Exam consultation

The exam and midterm consultation will be held in the joint consultation organized by CePro on July 18th.

 Chapter 1: Lebesgue measure
The problem of measure
Additivity, subadditivity, countable additivity
Outer measure
Cantor set

[Tao, Chapter 7, 7.1, 7.2], [Dacorogna, 15.5] *

*For all future weeks, the indication of chapters and sections is just to orient yourself and may not contain all/may contain more material. The material to be learnt is the one discussed in class.

Measurable sets
Measurable functions

[Tao's book 7.4,7.5]

Lebesgue integral of signed functions
Dominated convergence theorem

Fourier series, Fourier and Plancherel theorems

[Tao, 5.5]

Easter week

Uniform convergence of Fourier series
Fourier series in sin or cos only

Chapter 4: Fourier transform
The Fourier transform

[Dacorogna, 17.7, 17.8, 18.1, 18.2]

The heat equation on a finite interval
The Laplace equation in a box

[Dacorogna, 19.1, 19.2] [Stein, Fourier Analysis, Ch. 5.2.3]

The wave equation: derivation, D’Alembert solution, Fourier transform solution, solution on an interval

[Dacorogna, 19.3.1, 19.4]

We organise a Q&A session with the teaching assistants on Friday, 18.06.2021 from 9-11 am where you can ask questions both on the theory and on the exercises. The format of the Q&A session will still be hybrid:

•  on the EPFL campus in room MED 0 1418 (in accordance with the newest EPFL directives).
  
• zoom-room of the exercise class (Meeting ID: 811 2240 6103, Passcode: 989995).

Before and after the Q&A session it will be possible to ask questions via Piazza.