| This thesis concerns the formulation and derivation of a generalization of a collection of basis functions originally devised by Norbert Wiener for function approximation over the entire real line. The generalized basis set may be parameterized by the polynomial rate of decay of the basis functions at infinity.;In order to explore the possible utility of the generalized basis set, we first investigate the applicability of the fast Fourier transform algorithm to Jacobi polynomial expansions. We show that such applicability is robust (efficient and accurate) for certain classes of Jacobi polynomials. In addition, we explore the extent to which Jacobi-Gauss-type nodal sets serve as Lebesgue-optimal interpolation sets. We extend our results to two dimensional triangular simplices to obtain the best-known Lebesgue constants to date to the author's knowledge.;Wiener's generalized basis over the infinite interval is a direct mapping of a generalized Fourier series over the finite interval. Using the properties of Jacobi polynomials and the generalized Fourier series, we are able to show that the generalized Wiener basis set is L2 orthonormal for any choice of the decay parameter. In addition, we show various other useful properties including fast Fourier transform applicability, efficient decay parameter modification, and sparsity and spectral properties of the stiffness matrix.;We conclude our investigation with a few examples pertaining to function approximation and solutions to partial differential equations. Although we do not claim to have developed a panacea for spectral expansions on infinite intervals, we present the generalized Wiener basis set as a strong competitor to existing methods. |
