Lecturer

Francesca Da Lio

Coordinator

Thomas Stucker

Content

This course will cover the following topics

• Real and complex Hilbert spaces, Hilbert bases and Riesz representation Theorem.
• Fourier series of a function in \(L^2([-\pi, \pi]; \mathbb{C})\), relationship between the regularity of a function and the asymptotic behavior of the Fourier coefficients. Application to the resolution of linear partial differential equations with various boundary conditions in \([-\pi, \pi]\).
• Fourier Transform in \(\mathbb{R}^d\) and its elementary properties, relationship between the regularity of the function and the asymptotic behaviour of its Fourier transform, relationship between the summability of the function and the regularity of it Fourier transform. Application to the resolution of linear partial differential equations with various decay conditions in \(\mathbb{R}^d\).
• Compact operators on Hilbert spaces, self-adjoint operators, the spectral theorem and applications to eigenvalue problems.

You can find more information about the course in the Course Catalogue.

Lectures

Day	Time	Room
Thursday	13:15-14:00	HG E 1.2
Friday	10:15-12:00	HG G 3

The first lecture will take place on Thursday February 20.

The diary of the lectures and the class notes are available HERE.

In some of the lectures the EduApp will be used. Please install it here: EduApp.

Exercises

A new exercise sheet will be posted here each Friday starting February 21. The exercises will be related to topics covered in the lectures that week. We encourage you to look at the problems and prepare questions for the exercise class on Wednesday. The deadline for handing in the problem set is at the beginning of the exercise class the following Wednesday. After the exercise class solutions will be posted here.

Starting from Problem Set 1, you will find a bonus exercise in every Problem Set. Typically, it will be an exam-level multiple choice question. Based on the number of correctly solved bonus exercises, you can earn a grade bonus of up to 0.125 (to cumulate with your grade bonus from Analysis III) for your final grade.

Exercises marked with a ♣ sign are for the bonus; exercises marked with a ★ sign are more challenging.

Things to keep in mind:

• With the exeption of the bonus exercises, handing in the solutions gives no points for the final exam, although working on the problem sets will greatly increase your chances to pass the exam.
• The purpose of giving you problem sets is to help you learn the material and develop your problem solving skills.
• The purpose of giving you the solutions and discussing them in exercise class is to show you how to solve the problems you could not solve or to show you more efficient proofs.
• The purpose of letting you hand in the solutions and us correcting them is not to assess your problem solving skills, but to give you feedback on how you write mathematics.

Exercise Classes

The first exercise class will take place on Wednesday February 26.

You can enroll for an exercise class on myStudies.

Exam Preparation

• In the bachelor's program in mathematics, according to the new regulations of 2021, the course unit Analysis III is examined together with Analysis IV. The written exam lasts 180 minutes and will take place in the exam session of summer 2025 (alternatively winter 2026).
• The exam program from Spring Semester 2024 is available HERE.
• The table below provides a list of exercises for exam preparation, along with past exam questions. The solution will be uploaded on Polybox

Recommended Literature

• [Iac] Lecture Notes, by Mikaela Iacobelli.
• [AmDaMe] "Introduction to measure Theory and Integration" by Ambrosio, Da Prato and Mennucci (Edizioni della Normale, 2011).
• [Bo] "Méthodes mathématiques pour les sciences physiques" by Bony (École polytechnique, 2000).
• [CaDA] "Introduction to Measure Theory and Functional Analysis" by Cannarsa and D'Aprile (Springer, 2008).
• [Co] "A Course in Functional Analysis" by Conway (Springer, 2007).
• [Ev] "Partial Differential Equations" by Evans (American Mathematical Society, 2010).
• [Kr] "Introductory Functional Analysis with Applications" by Kreyszig (John Wiley & Sons, 1978).
• [StSh] "Fourier Analysis: an Introduction" by Stein and Shakarchi (Princeton University Press, 2003)
• [Ci] "Exercises and Problems in Mathematical Methods of Physics" by Cicogna (Springer, 2020)