GradTopicsSpring2025

Signal recovery and harmonic analysis, Spring 2025

Course: Math 578 Topics in Harmonic Analysis
Instructor: Alex Iosevich
Meeting time: Mondays 6.00-8.30 p.m. 1106A
Grading: Based on in-class presentations and attendance

Syllabus:

-Signal Recovery:

The basic question here is the following. Let $f:{\mathbb Z}_N^d \to {\mathbb C}$ . Let $\widehat{f}(m)=N^{-\frac{d}{2}} \sum_{x \in {\mathbb Z}_N^d} \chi(-x \cdot m) f(x),$ where $S \subset {\mathbb Z}_N^d.$ Is it possible to recover the values of

f(x) exa

We shall implement several of these mechanisms using Python. We shall develop a variety of analytic tools to study this problem, and this is where we now turn our attention.

- Bourgain's

Λq problem: \Lambda_q p

One of the tools we are going to use to study signal recovery is a celebrated theorem due to Jean Bourgain (1989). In this context, he proved that if

S \subset {\mathbb Z}_N^d, |S|=\lceil N^{\frac{2d}{q}} \rceil, q>2,

{\left( \frac{1}{N^d} \sum_x {|f(x)|}^q \right)}^{\frac{1}{q}} \leq C(q) {\left( \frac{1}{N^d} \sum_x {|f(x)|}^2 \right)}^{\frac{1}{2}},

provided the Fourier transform of f vanishes outside of S and S is suitably generic. The proof uses some very clever and intricate probabilistic tools.

- Restriction theory:

Bourgain's result above is an example of a class of important problems in analysis called restriction/extension inequalities for the Fourier transform. We are going to develop some basic skills in analyzing such inequalities in a discrete setting, Euclidean space, and, if time permits, on Riemannian manifolds.

- Real-life applications:

Towards the end of the course, we are going to combine the methods of signal recovery and the reinforcement learning techniques to impute missing values in time series, study inventory problems in data science, and similar questions.