On the Elementary Symmetric Polynomials, Inverse Vandermonde Matrix, and its Application in Signals and Images

The Vandermonde matrix naturally appears in systems of equations for applications of polynomial approximation. Solving the system, which requires inverting the Vandermonde matrix, may be preferred because researchers can utilize efficient matrix computation methods. However, the Vandermonde is ill-conditioned and prevents standard methods from calculating accurate entries to the inverse. Many of the derived algorithmic solutions to the Vandermonde inverse contain a mathematical expression called the Elementary Symmetric Polynomial (ESP). The ESP is computationally inefficient to solve directly, so there are also algorithms to solve them. The state-of-the-art ESP and inverse methods are fully recursive and may be more susceptible to noise perturbations in the Vandermonde matrix than a closed-form inverse method. In addition, the performances of the methods in a variety of applications in polynomial approximation are unexplored. This thesis investigates the utility of the state-of-the-art methods against a novel ESP and closed-form Vandermonde inverse method in applications of polynomial interpolation and Fourier reconstruction in the presence of Gaussian noise.M.A.S