Gibbs phenomenon
E898519
The Gibbs phenomenon is the persistent overshoot and oscillation that occurs near jump discontinuities when approximating a function with its Fourier series or other truncated series expansions.
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
effect in Fourier analysis
ⓘ
mathematical phenomenon ⓘ |
| alsoKnownAs | Gibbs–Wilbraham phenomenon NERFINISHED ⓘ |
| appearsIn |
approximation of sawtooth waves by Fourier series
ⓘ
approximation of square waves by Fourier series ⓘ approximation of step functions by Fourier series ⓘ |
| cause |
pointwise convergence failure at discontinuities
ⓘ
ringing artifacts in reconstructed images ⓘ ringing artifacts in reconstructed signals ⓘ |
| characterizedBy |
nonuniform convergence of Fourier series near jumps
ⓘ
oscillations localized near discontinuities ⓘ persistent overshoot that does not vanish as more terms are added ⓘ |
| concerns | behavior of partial sums of series near discontinuities ⓘ |
| describes |
oscillations near jump discontinuities in truncated series expansions
ⓘ
overshoot near jump discontinuities in Fourier series approximations ⓘ |
| field |
Fourier analysis
NERFINISHED
ⓘ
approximation theory ⓘ mathematical analysis ⓘ signal processing ⓘ |
| firstAnalyzedBy | Henry Wilbraham GENERATED ⓘ |
| hasApproximateOvershoot | about 9 percent of the jump size ⓘ |
| hasConsequence |
Fourier series cannot converge uniformly at jump discontinuities
ⓘ
approximation error near jumps does not go to zero pointwise ⓘ audible ringing near sharp transients in audio signals ⓘ visual ringing near edges in images ⓘ |
| mitigatedBy |
Cesàro summation
NERFINISHED
ⓘ
Fejér summation NERFINISHED ⓘ Gegenbauer reconstruction methods NERFINISHED ⓘ smoothing filters in the frequency domain ⓘ spectral viscosity methods ⓘ windowing techniques ⓘ |
| namedAfter | Josiah Willard Gibbs NERFINISHED ⓘ |
| occursIn |
Fourier series of piecewise smooth functions with jump discontinuities
ⓘ
digital signal processing ⓘ image processing ⓘ spectral approximations of discontinuous solutions ⓘ spectral methods for partial differential equations ⓘ truncated Fourier series ⓘ truncated orthogonal series expansions ⓘ |
| overshootLimit | approximately 0.08949 times the jump magnitude ⓘ |
| persistsUnder |
increasing number of Fourier series terms
ⓘ
partial sums of Fourier series ⓘ |
| relatedTo |
Dirichlet kernel
NERFINISHED
ⓘ
Fejér kernel NERFINISHED ⓘ Runge phenomenon ⓘ pointwise convergence of Fourier series ⓘ spectral ringing ⓘ uniform convergence of series ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.