---
tag: VMF169AT67
kind: concept
name: Gibbs phenomenon
msc: [42A20, 42A16]
sources:
  - url: https://en.wikipedia.org/wiki/Gibbs_phenomenon
same_as:
  wikidata: Q5374
  wikipedia: Gibbs phenomenon
---

## Summary

The Gibbs phenomenon is the overshoot of the partial sums of a [[Fourier series|G12KXDHPPS]] near a [[jump discontinuity|004FH91BN5]] of the function.
As more terms are added, the overshoot moves closer to the jump but does not shrink.
It tends to about $9\%$ of the size of the jump on each side.
It shows that the [[convergence of a Fourier series|V6Q24CHJG8]] at a jump cannot be uniform.
It matters in signal processing, where truncating frequencies causes ringing near sharp edges.

## Explanations

### L1 human

Try to draw a step with a finite number of smooth waves.
The flat parts come out nearly right, but just before and after the step the curve shoots past the target and wiggles.
With more waves the wiggles get narrower and crowd towards the step, yet the first spike stays about $9\%$ of the step height too high.
For the square wave that jumps from $-1$ to $1$, the partial sums peak near $1.179$ instead of $1$.