The first chapter is based on applying the Poisson summation formula to a constrained optimization problem. Motivated by Shannon sampling theorem and results on shift-invariant subspaces, we establish a compatible framework for the two key factors: the accuracy constraint, which is described in the frequency space, and the efficiency function, which is expressed in the regular space. We derive the optimal wavelet, denoted as the double-sinc function, that obtains the smallest support while remaining first order accuracy. Based on this wavelet, we further improve its accuracy by loosen up the constraint in support and manage to achieve nearly optimal efficiency.

The goal of the second chapter is to recover the underlying signal from its superposed randomly-shifted noisy measurement, motivated by multi-reference alignment and Cryo-EM problem. The general setting is that we observe samples from noisy signal that is acted by a random group action, and we would like eliminate those noises, one type at a time. In our particular setting, rational Fourier monomials and total Fourier product are invariants under the group action and hence partially remove the effect of the noise from the group action. We then apply central limit theorem to eliminate the Gaussian noise. Finally, we apply the split Bregman algorithm in compressed sensing to obtain an explicit solution assuming that the signal is compactly supported.

The third chapter is dedicated to applying a variation principle to the Euler-Poisson equation for periodic flow in a diode to optimize the flux difference, which could potentially exceed the Child-Langmuir limit. We derive a set of dual equations and boundary conditions and use upwind method to solve both the forward and backward equations. In our numerical experiment we derive a periodic solution whose flux goes above the CL limit before the physical setting or the method of characteristics breaks down.