The Gibbs Phenomenon and its Resolution

Sammanfattning: It is well known that given an arbitrary continuous and periodic function f(x), it is possible to represent it as a Fourier series. However, attempting to approximate a discontinuous or non periodic function, using a Fourier series, yields very poor results. Large oscillations and overshoots appear around the points of discontinuity. Regardless of the number of terms that are included in the series, these overshoots do not disappear, they simply move closer to the point of discontinuity. This is known as the Gibbs phenomenon. In the 1990's David Gottlieb and Chi-Wang Shu introduced a new method, entitled the Gegenbauer procedure, which completely removes the Gibbs phenomenon. We will review their method, as well as present a number of examples to illustrate its effect. Going one step further we will discover that not all orthogonal polynomials may be treated equal in terms of this Gegenbauer procedure. When replacing Gegenbauer polynomials with Chebyshev or Legendre polynomials, it appears as though the inability to vary ultimately makes them ineffective.