Chasing traces of analyticity with applications to operator theory
The project concerns questions in Fourier analysis related to spectral structure of functions with various unilateral decay conditions. The research is motivated by applications in theory of linear operators and other parts of mathematics.
Start
2025-08-01
Planned completion
2029-08-01
Main financing
Research group
Project manager at MDU
In the early 19th century, the Frenchman Joseph Fourier had the brilliant, but at the time very controversial, idea that every mathematical function should be able to be constructed as an infinite sum of sine waves. His French colleagues were skeptical, but over time it turned out that Fourier's intuition was correct. The theory that deals with the phenomenon is now called Fourier analysis, and the sum of sine waves that reconstructs the function is called the Fourier series of the function, or it's the Fourier transform.
Fourier analysis is a strikingly successful theory that continues to amaze with its wide-ranging applications in mathematics, science, and engineering. In signal processing, one is often interested in filtering a signal, for example with the aim of removing unnecessary noise, and such an operation can be implemented mathematically by discarding high-frequency sine waves from the Fourier series of the function. In quantum physics, Fourier analysis is used to mathematically formalize Heisenberg's well-known uncertainty relations, which postulate universal limits on the accuracy of certain types of physical measurements. Recently, new Fourier-analytic reconstruction formulas have played a central role in solving the sphere packing problem, which aims to find optimal ways to pack multidimensional spherical particles.
Project objective
My research seeks to further develop Fourier analysis. My questions fall under the theme of the uncertainty principle in Fourier analysis, the name inspired by Heisenberg's uncertainty relation. I am interested in determining the structure of the Fourier transform of a function that is a consequence of certain local requirements on the decay of the function's values. In particular, I want to find the limit on how spread out the frequencies can be under various assumptions. The questions are inspired by unsolved problems in several other mathematical fields, such as operator theory, but the project has as an ambition to develop central Fourier-analytic tools that can be applied more generally.
