Topics on Stability, Interpolation, and Expansions

This research project focuses on solving key problems at the intersection of two areas of mathematics: functional analysis, the study of functions and function spaces, and harmonic analysis, which analyzes phenomena of a periodically recurrent nature. Both subjects have important applications in various fields, among which are physics, engineering, and computer science. Our work will explore three central topics, each of which could lead to valuable insights and new ways to tackle practical challenges. The first goal is to study a type of mathematical structure known as operator semigroups, which help describe how certain systems or processes evolve over time. Such objects appear practically in every field using mathematics, modeling things such as the distribution of heat, the spread of information in networks, or the evolution of stock markets. We aim to develop new methods for understanding how stable and predictable these systems are over the long term, both in qualitative and quantitative terms. Specifically, our goal is to find ways to detect whether a process, except for some singularities, dies out, and if so, at which rate. Suppose one has a discrete collection of points on the complex plane and that for each point, one has an associated value adhering to certain growth bounds. In the second part of this project, we aim to determine conditions to guarantee the existence of an entire function that matches these values at the respective points while reflecting the growth bounds across the entire plane. Current methods often rely on restrictive conditions, and we aim to make this process more adaptable. Such results could then be applied in harmonic analysis, in particular when breaking down complex signals or functions into discrete parts. These techniques are essential from a practical point of view, for instance, in data compression or image processing. The final objective is to develop novel discrete ways to detect when functions exhibit nearly optimal decay in time and frequency. Such signals, whose information is primarily focused in a finite time interval and which behave very regularly, often appear naturally in quantum mechanics. As a result, our findings will lend themselves naturally to a variety of problems in that field, for instance, when looking at the solvability of the Schrödinger equation. Additionally, our framework is closely related to the existence and optimality of Gabor frames, which decompose signals into discrete values.