A Review of David Gottlieb’s Work on the Resolution of the Gibbs Phenomenon

Given a piecewise smooth function, it is possible to construct a global expansion in some complete orthogonal basis, such as the Fourier basis. However, the local discontinuities of the function will destroy the convergence of global approximations, even in regions for which the underlying function is analytic. The global expansions are contaminated by the presence of a local discontinuity, and the result is that the partial sums are oscillatory and feature non-uniform convergence. This characteristic behavior is called the Gibbs phenomenon. However, David Gottlieb and Chi-Wang Shu showed that these slowly and non-uniformly convergent global approximations retain within them high order information which can be recovered with suitable postprocessing. In this paper we review the history of the Gibbs phenomenon and the story of its resolution.

Identification of Nonlinear Systems by Haar Wavelet

This study addresses the identification of autonomous nonlinear systems. It is assumed that the function form in the nonlinear system is known, leaving some unknown parameters to be estimated. It is also assumed that the free responses of the system can be measured. Since Haar wavelets can form a complete orthogonal basis for the appropriate function space, they are used to expand all signals. In doing so, the state equation can be transformed into a set of algebraic equations in unknown parameters. The technique of Kronecker product is utilized to simplify the expressions of the associated algebraic equations. Together with the least square method, the unknown system parameters are estimated. Several simulation examples verify the analysis.