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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Gibbs phenomenon canonical | 4 |
| Gibbs–Wilbraham phenomenon | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10992353 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Gibbs phenomenon Context triple: [Dirichlet conditions, relatedTo, Gibbs phenomenon]
-
A.
Runge phenomenon
The Runge phenomenon is a numerical analysis effect where high-degree polynomial interpolation, especially at equally spaced points, produces large oscillations and poor approximations near the interval endpoints.
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B.
Du Bois-Reymond function
The Du Bois-Reymond function is a classic example of a continuous but nowhere differentiable function, illustrating pathological behavior in real analysis.
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C.
Riemann–Lebesgue lemma
The Riemann–Lebesgue lemma is a fundamental result in Fourier analysis stating that the Fourier coefficients (or transform) of an integrable function vanish at infinity.
-
D.
Fourier series
A Fourier series is a way of representing a periodic function as an infinite sum of sines and cosines with appropriately chosen coefficients.
-
E.
Fresnel integrals
Fresnel integrals are special functions in mathematics that describe the complex oscillatory behavior of wave diffraction and interference, particularly in optics.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Gibbs phenomenon Target entity description: 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.
-
A.
Runge phenomenon
The Runge phenomenon is a numerical analysis effect where high-degree polynomial interpolation, especially at equally spaced points, produces large oscillations and poor approximations near the interval endpoints.
-
B.
Du Bois-Reymond function
The Du Bois-Reymond function is a classic example of a continuous but nowhere differentiable function, illustrating pathological behavior in real analysis.
-
C.
Riemann–Lebesgue lemma
The Riemann–Lebesgue lemma is a fundamental result in Fourier analysis stating that the Fourier coefficients (or transform) of an integrable function vanish at infinity.
-
D.
Fourier series
A Fourier series is a way of representing a periodic function as an infinite sum of sines and cosines with appropriately chosen coefficients.
-
E.
Fresnel integrals
Fresnel integrals are special functions in mathematics that describe the complex oscillatory behavior of wave diffraction and interference, particularly in optics.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
effect in Fourier analysis
ⓘ
mathematical phenomenon ⓘ |
| alsoKnownAs |
Gibbs–Wilbraham phenomenon
ⓘ
linked to:
Gibbs phenomenon
|
| 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
ⓘ
approximation theory ⓘ mathematical analysis ⓘ signal processing ⓘ |
| firstAnalyzedBy | Henry Wilbraham ⓘ |
| 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
ⓘ
Fejér summation ⓘ
linked to:
Fejér kernel
Gegenbauer reconstruction methods ⓘ
linked to:
Gegenbauer polynomials
smoothing filters in the frequency domain ⓘ spectral viscosity methods ⓘ windowing techniques ⓘ |
| namedAfter | Josiah Willard Gibbs ⓘ |
| 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
ⓘ
Fejér kernel ⓘ Runge phenomenon ⓘ pointwise convergence of Fourier series ⓘ spectral ringing ⓘ uniform convergence of series ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Gibbs phenomenon Description of subject: 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.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Gibbs phenomenon