Abstract: In this thesis, we study topics related to the application of harmonic analysis to data science. In Chapter 2 we look at applications to signal recovery. Given a function $f:\mathbb{Z}_N \to \mathbb{C}$ (the ``signal'') such that for some $M \subset \mathbb{Z}_N$, $\{f(x)\}_{x \in M}$ is unknown, we study under what conditions the signal can be (approximately) recovered and mechanisms for said recovery. In this chapter, we study this with respect to approximate sparsity of the Fourier transform of $f$.

In Chapter3 we study this with respect to a generalization of the Fourier ratio. In particular, we generalize the Fourier ratio with respect to arbitrary orthonormal bases. We also connect the Fourier ratio to Kolmogorov complexity.

Finally, in Chapter 4, we look at applications of fractal geometry to dimensionality detection. We show that the discrete Riesz energy can be used to estimate the continuous Riesz energy of a measure, $\mu$, given a sequence of points, $\{x_{i}\}_{i = 1}^{\infty}$, where the points, in certain ways, increasingly represent $\mu$. We will also apply to this to extend connections between Erd\H{o}s and Falconer distance type problems.