Representation of Functions in Basis Sets

Abstract

The orthogonal basis sets most often used in spectral methods are the Chebyshev and Legendre polynomials on a bounded domain, or a Fourier basis set for periodic functions. We discuss in this chapter the expansions of Gaussian and Kappa distributions of kinetic theory in Hermite and Laguerre polynomials on the infinite and semi-infinite intervals, respectively. The spectral convergence properties of these expansions is demonstrated numerically and analytically. The expansions of \(\sin (x)\) in Hermite polynomials, and of the Maxwellian distribution in Chebyshev polynomials are also considered. The basic principles of Fourier series are presented and applied to quantum mechanical wave packets as well as the analysis of free induction decay signals. The resolution of the Gibbs phenomenon with the Gegenbauer reconstruction method is compared with the inverse polynomial reconstruction method. A resolution of the Runge phenonmena is also presented.