I propose topics for Bachelor’s and Master’s theses in the areas of harmonic analysis, Fourier analysis, and time–frequency analysis. If you are interested, please contact me via email
Sampling in spaces of analytic functions
Sampling theory addresses the following fundamental question: under what conditions on a function (signal) \(f\) and on a set \(\Lambda\) can the function be uniquely reconstructed from its samples \(\{f(\lambda)\}_{\lambda \in \Lambda}\)?
On the signal side, one typically assumes that \(f\) belongs to a prescribed function space, such as Paley–Wiener spaces, Bergman spaces, Bargmann–Fock spaces, or (quasi) shift-invariant spaces. On the other hand, the answer depends crucially on the density and structure of the sampling set \(\Lambda\).
The subject includes a number of classical results that can serve as starting points for student projects, as well as many accessible open problems suitable for further research.
- Olevskii, A., Ulanovskii, A. Functions with Disconnected Spectrum: Sampling, Interpolation, Translates. AMS, University Lecture Series, 65, 2016.
Crystalline measures, tilings and stability of shifts
A (tempered) measure \(\mu\) on \(\mathbb{R}^d\) is called crystalline if both \(\mu\) and its Fourier transform \(\widehat{\mu}\) are discrete measures, that is, \[ \mu=\sum_{\gamma\in\Gamma} c_\gamma \delta_\gamma, \qquad \widehat{\mu}=\sum_{s\in S} d_s \delta_s, \] where \(\Gamma\) and \(S\) are discrete sets of points. Typical examples arise from periodic structures such as lattices, but more complicated nonperiodic configurations also exist.
In this project we will study basic properties of crystalline measures and their connections to several questions in harmonic analysis, including sampling and interpolation problems, stability of systems of shifts, and tiling problems.
- N. Lev and A. Olevskii. A. Quasicrystals and Poisson's summation formula. Invent. Math., 200(2):585–606, 2015.
- B. Matei and Y. Meyer. Simple quasicrystals are sets of stable sampling. Complex Var. Elliptic Equ., 55(8-10):947–964, 2010.
Recommended background. It is helpful (though not strictly required) to have taken one or more of the following courses prior to starting the project or Master’s thesis:
- TMA4170 Fourier Analysis,
- TMA4175 Complex Analysis,
- TMA4230 Functional Analysis.
I am also happy to discuss thesis topics in any area of harmonic analysis, Fourier analysis, or complex analysis, and I welcome suggestions if you have your own ideas for a project.