(Georgia Institute of Technology, 2023-04-27)
Bailey, Victor
Dynamical Sampling is a superordinate term classifying the set of inverse problems arising from considering samples of a signal and its future states under the iterative action of a linear operator. A central problem in this area is the following: let $f \in \ell^2(I)$ where $I=\{1, \ldots, d\}$. Suppose for $\Omega \subset I$ we know $\{{ A^j f(i)} : j= 0, \ldots l_i, i\in \Omega \}$ for some operator $A: \ell^2(I) \to \ell^2(I)$. What are conditions on $\Omega, A$, and $l_i$ that allow the stable reconstruction of $f$? In this thesis we apply the Dynamical Sampling framework to obtain results regarding reconstructing Paley-Wiener spaces of finite combinatorial graphs.
In separable infinite-dimensional Hilbert spaces, the Dynamical Sampling problem is typically posed the following way: what are necessary and sufficient conditions so that $\{T^n \varphi \}_{n \geq 0}$ is a frame (where $T \in B(H)$ and $\varphi \in H$)? Extending the results in this area by applying tools from the theory of shift-invariant subspaces of the Hardy Space on the bidisc, in this thesis we present necessary and sufficient conditions for systems of the form $\{T_1^iT_2^jf_0\}_{i,j \geq 0}$, where $T_1, T_2 \in B(H)$ commute, to be a frame.