Lecture courses/details

Lectures

Timeslot:
13:00 - 14:45

Room:
Y27H46 Seats: 40

>>Course Intro Presentation

Review of topics

Week 1: Introduction/Historical Background. Fourier series on the torus. Basic properties of Fourier coefficients (Riemann-Lebesgue lemma). Partial sums and the Dirichlet kernel. Criteria for pointwise convergence (Dini).
Week 2: Partial sums and continuity (duBois-Reymond). Summability methods: Fejér and Poisson kernels. Approximate identities. Density of trigonometric polynomials in L^p.
Week 3: Norm convergence of partial sums: L^2 setting. Relation to boundedness of partial summation. The Maximal function: introduction and basic properties.
Week 4: Weak L^1 and L^p boundedness of the maximal function
Week 5: The maximal function and a.e. convergence of radially bounded approximate identities
Week 6: Weak L^p spaces, maximal operators and a.e. convergence, Marcinkiewicz interpolation
Week 7: Fourier transform, Schwartz space
Week 8: Tempered distributions and their Fourier transform
Week 9: Hilbert transform

VVZ Link

Exercises

13:00 - 14:45

Y27H46
Seats: 40

Exercises Harmonic Analysis
Tutor: Stefano Rossi

Online Exercise submission

Each lecture will be accompanied by a problem set that revisits and expands on a selection of topics from the lecture. These exercises will be discussed in the exercise session on Wednesdays, according to the following two-week rhythm:

The context of each exercise, and possibly hints towards its solution, are given.
You have one week to work on solutions and hand them in.
Another week later, solutions are discussed in the exercise session.

While you are not required to hand in solutions, you are encouraged to do so. Irrespective of this, you should present (at least) one problem and its solution in an exercise session.

Downloads

Problem Set 1
Problem Set 2
Problem Set 3
Problem Set 4
Problem Set 5
Problem Set 6
Problem Set 7
Problem Set 8
Problem Set 9

Remarks on the gliding hump

FAQ

Details

Overview

Harmonic Analysis originates with representations of functions as the superposition of basic "waves". In this course we develop aspects of the corresponding real-variable theory: Fourier series and transforms, singular integrals and further topics. These ideas and techniques have become a powerful tool in many branches of mathematics and applications.

Topics will include: Fourier series on the circle and their convergence, summability methods, Fourier transforms, maximal functions and L^p interpolation, the Hilbert transform, singular integrals (of Calderón-Zygmund type).

Literature

The main references for this course are the two books

"Fourier Analysis" by J. Duoandikoetxea
"Classical and multilinear harmonic analysis: Volume 1" by C. Muscalu and W. Schlag

Futher book recommendations are

"Fourier Analysis" and "Functional Analysis" by E.M. Stein and R. Shakarchi
"Classical Fourier Analysis" by L. Grafakos

Exam

The exam will be oral, 30 minutes duration. Topics include all material from the lecture and the exercises. Active participation in the exercise session is thus highly encouraged.

For further information please contact:

Exam

Exam dates: missing

Institute of Mathematics

Lecture courses/details

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