##  [Gibbs Phenomenon](/gibbs-phenomenon-0) 

 Definition

The persistent overshoot and oscillatory ringing near a jump discontinuity that occurs when approximating a piecewise‑smooth function by truncated Fourier series or other global spectral expansions; the overshoot does not vanish as the number of terms grows, although it narrows.

 

 

 

 

 

 





## Principle

Principle

Global spectral approximants enforce smooth basis behavior across a domain and cannot simultaneously match jump discontinuities and uniform convergence; partial sums concentrate oscillations near discontinuities because of slow decay of high‑frequency coefficients.

 

 

 

 

 





## Demonstration

Demonstration

Approximate a 2π‑periodic sawtooth function with its N‑term Fourier partial sum: near each jump the partial sum overshoots the limiting value by approximately 9% of the jump magnitude, and the oscillations become more localized but their amplitude persists as N→∞.

 

 

 

 

## Misapplication

Misapplication

Expecting uniform convergence at points of discontinuity or attributing the overshoot to numerical rounding: Gibbs phenomenon is analytic and persists in exact arithmetic for spectral truncations.

 

 

 

 

 





## Consequence

Consequence

Recognition of the phenomenon motivates use of filters, mollifiers, localized bases, or nonlinear reconstruction schemes to reduce ringing and improve approximation near discontinuities.

 

 

 

 

## Reversal

Reversal

For functions that are continuously differentiable across the domain, spectral partial sums converge uniformly and no Gibbs overshoot occurs; thus smoothness across the domain is the reversal condition.

 

 

 

 

 





## Boundary

Boundary

Occurs for global spectral approximations of functions with jump discontinuities or sharp gradients; it does not describe behaviour of local approximation schemes (finite elements with local refinement) which can avoid global ringing.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Tension exists with 'ringing' seen in signal processing where discrete sampling and windowing produce similar artifacts; Gibbs phenomenon is the mathematical continuous‑time spectral origin of such ringing but distinct from sampling aliasing.

 

 

 

 

 





## Synthesis

Synthesis

Gibbs phenomenon is the nonvanishing overshoot and localized oscillation produced by truncating global spectral expansions at discontinuities, reflecting a fundamental incompatibility between global smooth bases and abrupt jumps.