Gibbsjelenség

Gibbsjelenség, also known as the Gibbs phenomenon, is a term used in mathematics to describe the behavior of Fourier series approximations of a function that has a jump discontinuity. When a Fourier series is used to approximate a function with a jump discontinuity, the approximation exhibits an overshoot or undershoot near the discontinuity. This overshoot does not diminish as more terms are added to the Fourier series; instead, it oscillates with increasing frequency and decreasing amplitude as the approximation gets closer to the discontinuity.

The Gibbs phenomenon is named after the physicist Josiah Willard Gibbs. He observed this behavior in the context of approximating a square wave with a Fourier series. The overshoot at the discontinuities is approximately 9% of the jump in function value. While the Fourier series converges pointwise to the function everywhere except at the discontinuity, the presence of the Gibbs phenomenon means that the convergence is not uniform across the entire domain, particularly around the jump. This behavior is characteristic of the convergence of Fourier series and other orthogonal series expansions when approximating functions with discontinuities. It highlights a limitation of pointwise convergence in representing sharp changes in functions.