Department of Mathematics, Indian Institute of Science, Bangalore 560 012
Fourier analysis and applications (Aug-Dec 2021)
Mon, Wed, Fri 12:00-1:00, Online (MS Teams)
Description: We introduce the basics of Fourier analysis and primarily concentrate on the groups ℤ/(n), (ℤ/(2))n, ℤd and ℝd. After covering the basic theory (either separately or under the banner of locally compact abelian groups), including duality, Plancherel's theorem and Poisson summation etc., we focus on many applications within and outside mathematics. A representative list is below - it is exceedingly unlikely that there will be time for all of the following topics!
Prerequisites: It is easiest to cover the theory if one knows measure theory and functional analysis (mainly understand Lp spaces, in particular p=2). But a major part of the course will only require basic analysis and group theory. Hence, depending on the audience, I may assume knowledge of Lebesgue integration theory and Hilbert spaces, or try to fill it in during the course (and omit one or two other topics). However, a certain amount of mathematical maturity is required to appreciate the issues. Also required is interest in techniques of analysis and willingness to wander into the diverse fields in which applications take us. UG 4th year and Int. PhD. (Math) 2nd/3rd year students are perhaps most suited to take this course, but others are welcome too.
- Equidistribution theorems Weyl's theorem for irrational rotations
- Isoperimetric inequality (in a limited form)
- Quadratic reciprocity law
- Dirichlet's theorem on primes in an arithmetic progression
- Arrow's theorem and related questions in social choice, stability and noise sensitivity of Booelan functions
- Roth's theorem on three-term arithmetic progressions in dense sets of integers
- Wiener-Ikehara Tauberian theorem and a proof of the Prime number theorem
- Stability and sensitivity of Boolean functions
- Uncertainty principles Heisenberg's, Donoho-Stark, Benedicks', etc.
- Some applications to signal processing such as Shannon-Nyquist and 2WT theorem
- Additive combinatorics The Littlewood-Offord problem and its generalisations
- Erdös-Turán lemma on equidistribution and the distribution of zeros of polynomials
- Lattices Sphere packing problems. Crystallography and Bragg's law.
- Expander graphs - Margulis' construction
Grading: Yet to be decided convex combination of homework problems, mid-terms and final examinations, presentations.
Texts and other resources: My lecture notes are based on many sources. In no particular order (I may sample material from many places, including books not mentioned here).
- E. M. Stein and R. Shakarchi Fourier analysis: an introduction First part of the four volume The Princeton Lectures in Analysis, Princeton University Press, 2003.
- T.W. Korner Fourier analysis, Cambridge university press, 1989.
- R. O'Donnell Analysis of Boolean functions Cambridge university press, 2014.
- W. Rudin Fourier analysis on groups, Tata McGraw-Hill, 2006, 3rd ed.
- H. Dym and H. P. McKean Fourier series and integrals, Academic Press, 1985.
- M. Pinsky Introduction to Fourier analysis and wavelets