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.
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).
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
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.