Analysis IV (Fourier Theory and Hilbert Spaces) Spring 2024

Lecturer
Mikaela Iacobelli
Coordinator
Marco Badran

Content

This course will cover the following topics

You can find all the relevant informations about the course in the course catalog.

Time and place

Lectures take place in HG F3 on Wednesdays form 9:15 to 10:00 and on Fridays from 10:15 to 12:00. The first lecture will be on February 21st, 2024.

Exercise classes take place on Wednesday in different locations and different groups according to the table below. Please enroll to the exercise classes in MyStudies. The first exercise class will be on February 21st, 2024.

WhenWhereWhoLanguage
Wednesday 10:15-12:00HG G 26.1D. UrechGerman
Wednesday 10:15-12:00LEE D 105C. AmendEnglish or German
Wednesday 10:15-12:00ML F 40J. ConanEnglish
Wednesday 10:15-12:00ML H 43U. FaureEnglish
Wednesday 10:15-12:00ML J 34.1A. KünziEnglish or German
Wednesday 12:15-14:00ML F 40H. TöpelGerman
Office hours

Currently, there are no office hours, but they will be established later if necessary. If you wish to schedule a meeting, please contact the coordinator directly at marco.badran@math.ethz.ch for an appointment.

Diary of lectures

The diary will be updated each week with the topics covered in class.

DateTopic discussedNotes
Wednesday, February 21Introduction to the course, definition of (real and complex) vector spaces and inner product spaces. Examples: \(\mathbb{R}^d\) , \(\mathbb{C}^d\), \(\ell^2(\mathbb{C})\).
Friday, February 23 Examples of inner product spaces \(L^2(0,1)\), \(C([0,1])\) (examples 1.8 and 1.11). Norms, Cauchy-Schwarz inequality and triangular inequality with proof, Parallelogram identity, Pythagora’s theorem, Polarisation formulas, Ptolemy inequality. Lemma 1.24 (continuity of the inner product), exercise 1.2, example 1.27 (some norms on finite dimensional), example 1.28 (\(C([0,1])\) is an infinite dimensional vector space) and some classical norms.
Wednesday, February 28\(L^p\) spaces are normed spaces, some topological definitions: open ball, interior point, open set, closed set, convex set, topological vector space, Cauchy sequences, completeness. Comparison between norms on the same space, definition of Hilbert spaces, examples of Hilbert spaces (\(\mathbb{C}^d\), \(\ell^2\), \(L^2\))
Friday, March 1 A subspace of a Hilbert space is Hilbert if and only if it is closed; example of inner product space that is not complete: \((C([-1,1]),\|\cdot\|_{L^2})\), notion of density and closure of a set, separability, example of separable spaces, example of a non separable Hilbert space, basis of a Hilbert space, definition of orthonormal set for a Hilbert space. Statement of Bessel inequality and Parseval identity.
Wednesday, March 6Proof of the theorem about orthonormal sets of a Hilbert space (Bessel inequality, Parseval identity), Hilbert basis, Completeness criterion for separable Hilbert Spaces.
Friday, March 8Theorem of existence of an Hilbert basis, example of \(\ell^2\), and example of compactly supported sequences. Every separable Hilbert space of infinite dimension is isometric to \(\ell^2\), Hilbert bases are not algebraic bases, theorem about the existence of the projection on closed subspaces and characterisation of the orthogonal projection. Statement and proof of the theorem about the projection on closed convex set. Characterization of the projection onto a convex set, with proof.
Wednesday, March 13 Nontriviality of the complement of a proper closed subspace. Projection onto a subspace in terms of a Hilbert basis of the subspace. Every closed subspace has an orthogonal complement. Proposition about isometry of \(H\) and \(Y\oplus Y^\perp\) for any closed subspace \(Y\). Definition of linear and bounded operators, example of an unbounded operator (derivative).
Friday, March 15Definition of the norm of a linear operator and proof that it is equal to the Lipschitz constant. Continuity of linear operators if the dimension is finite. Example: the identity is not necessarily continuous if we change the topology. A linear operator between normed vector spaces is continuous if and only if it is bounded. Riesz' representation Theorem of continuous linear functionals on an Hilbert space, with proof. Corollary: canonical isometric isomorphism between a Hilbert space and its dual. Application: Von Neumann’s proof of the Radon-Nikodym theorem (not examinable).
Wednesday, March 20Motivation of the Fourier series, definiton of Fourier coefficients and Fourier partial sum. The Complex Stone Weierstrass Theorem (only statement). Theorem: the Fourier basis forms an Hilbert basis of \(L^2((-\pi,\pi);\mathbb{C})\) (proof using the Complex Stone Weierstrass).
Friday, March 22Consequences of the fact that the Fourier basis is an Hilbert basis: \(L^2\) convergence of partial sums, Parseval's identity, scalar product in terms of the Fourier coefficients. Example: Fourier series of sines and cosines. Proposition: characterization of real valued functions via their Fourier coefficients. Almost everywhere convergence of the Fourier partial sums, Carleson's Theorem (only statement). Examples: Foureir series of trigonometric polynomial, Fourier series of \(x\) and computation of the sum of inverse squares of the integers (Basel's problem)
Wednesday, March 27 Convergence criteria for series in Hilbert spaces. Example: regularity of limit functions. Fourier coefficients of the derivative. Asymptotic behaviour of Fourier coefficients of \(C^1\) functions.
Wednesday, April 10 Corollary about uniform convergence of the Fourier series of a \(C^1\) function. Fourier coefficients of higher derivatives of a function. Corollary: uniform convergence of the Fourier series of a function \(C^h\) with all its derivatives. Theorem: summability implies regularity (statement only).
Friday, April 12 Proof of the summability implies regularity theorem. Statement of the pointwise convergence theorem. Definition of Dirichlet kernel and properties (mean value and explicit expression). Riemann-Lebesgue lemma. Proof of pointwise convergence theorem. Introduction to the heat equation and sketch of the strategy to find solutions using Fourier series.
Wednesday, April 17
Friday, April 19
Wednesday, April 24
Friday, April 26
Friday, May 3
Wednesday, May 8
Friday, May 10
Wednesday, May 15
Friday, May 17
Wednesday, May 22
Friday, May 24
Wednesday, May 29
Friday, May 31

Exercises

A new exercise sheet will be posted here one week before each exercise class. The exercises will be related to topics covered in the previous week. The solutions will be posted by the end of the exercise class day.

Starting from Problem Set 1 you will find a bonus exercise in every Problem Set. Typically, it will be an exam-level multiple choice exercise. Those who will hand in at least 10 out of 12 correctly solved bonus exercise will get extra points in the final mark. The bonus is worth 0.125 points, to potentially cumulate with the Analysis III bonus, also worth 0.125, for an extra 0.25 in the final mark. The deadline for submitting the solutions to Problem Set and having the bonus exercise counted is Tuesday at midnight.

Things to keep in mind:

Problem set Due by Upload link Solutions
Problem set 0 February 28 Submission Solutions
Problem set 1 March 6 Submission Solutions
Problem set 2 March 12 Submission Solutions
Problem set 3 March 19 Submission Solutions
Problem set 4 March 26 Submission Solutions
Problem set 5 April 9 Submission Solutions
Problem set 6 April 16 Submission

Literature

You can find the lastest version of the course script here (password protected). Besides, below you find some textbooks that cover similar topics and that have been used to prepare lectures: