EHU Reading Course 2026 - Introduction to Fourier Analysis
EHU Reading Course 2026 - Introduction to Fourier Analysis
This reading course will serve as an introduction to Fourier Analysis aimed at students with a background in real and complex analysis, and lectures will be coordinated by
Mateus Sousa (BCAM Ramón y Cajal Fellow)
Daniel Eceizabarrena (BCAM Ramón y Cajal Fellow)
The course will be roughly divided into 3 parts:
I. The periodic setting (Fourier series);
II. The Euclidian setting (Fourier transform);
III. The finite group setting (Characters and the Fast Fourier transform).
The course will take place on classroom 0.12 of EHU Faculty of Sciences in Leioa, and will start on February 4th, 2026. The plan is to have a total of 13 sessions, divided into 10 lectures of 1h30 and 3 days for student presentations. Sessions will always take place on Wednesdays at 12:00. Student presentations will take place right after the last lectures of each part, and the distribution of topics will be done during the first meeting.
The course will follow roughly the contents presented in
[SS1] E. Stein, and R. Shakarchi - Fourier Analysis: An Introduction. Princeton University Press, Princeton, NJ 2003.
With some of the proofs being more in the spirit of
[SS3] E. Stein, and R. Shakarchi - Real Analysis: Measure Theory, Integration, and Hilbert Spaces. Princeton University Press, Princeton, NJ 2005.
Of course there are many other great references one can follow, such as:
J. Duoandikoetxea - Fourier Analysis. Graduate Studies in Mathematics, 29, AMS, Providence, RI 2001.
L. Grafakos - Classical Fourier Analysis. Graduate Texts in Mathematics, 249, Springer, New York, NY 2014.
C. Muscalu, and W. Schlag - Classical and Multilinear Harmonic Analysis. Cambridge University Press, New York, NY 2013.
There will 3 days of presentations, each with two talks of 30 minutes plus some time for questions.
The isoperimetric inequality
Based on section 4.1 of [SS1]
Weyl's equidistribution theorem
Based on section 4.2 of [SS1]
The time-dependent heat equation
Based on section 5.2.1 and 4.4 of [SS1]
The steady-state heat equation
Based on section 5.2.2 and 2.5.4 of [SS1]
Primes in Arithmetic progression (2 parts).
Based on sections 8.2 and 8.3 of [SS1]
Feb 4th: Motivation for Fourier Analysis