A comparison of error estimates at a point and on a set when solving ill-posed problems

The problem of correlating the error estimates at a point and on a correctness class is of interest to many mathematicians. Since the desired solution to a real ill-posed problem is unique, the error estimate obtained on the class becomes crude. In this paper, by assuming that an exact solution is a piecewise smooth function, we prove, for a special class of incorrect problems, that an error estimate at a point is an infinitely small quantity compared with an exact estimate on a correctness set.

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.