Topics in Harmonic Analysis: The HRT Conjecture and Sigma Delta Quantization

Abstract

In this thesis, we solve for special cases of the HRT Conjecture and providean overview of Σ∆-Quantization. The HRT conjecture has remained open ever since its introduction in 1996. Despite 25 years of concerted effort, even special cases of the conjecture with very stringent conditions are still unsolved. We solve special cases of the HRT conjecture for functions that are not smooth on a nonempty, bounded set and when the function has certain decay conditions. We demonstrate how non-smooth functions that do not satisfy the HRT conjecture cannot be non-smooth at only a finite number of points or a bounded set of points. We introduce new definitions to quantify both the size and orientation of a discontinuity and various techniques for the resulting calculations. We also study how a function’s failure to satisfy the HRT conjecture affects the function’s behavior at infinity. In order to do this, extensions of the real number line are defined along with new topologies on these extensions. These new topologies allow us to describe several new notions of convergence.