The main result of this thesis is an efficient protocol to determine the frequencies of a signal $C(t)= \sum_k |a_k|^2 e^{i ω_k t}$, which is given for a finite time, to a high degree of precision. Specifically, we develop a theorem that provides a fundamental precision guarantee. Additionally, we establish an approximation theory for spectral analysis through low-dimensional subspaces that can be applied to a wide range of problems. The signal processing routine relies on a symmetry between harmonic analysis and quantum mechanics. In this context, prolate spheroidal wave functions (PSWF) are identified as the optimal information processing basis. To establish rigorous precision guarantees, we extend the concentration properties of PSWFs to a supremum bound and an $\ell_2$ bound on their derivatives. The new bounds allow us to refine the truncation estimates for the prolate sampling formula. We also provide a new geometrical insight into the commutation relation between an integral operator and a differential operator, both of which have PSWFs as eigenfunctions.