Gibbs Phenomenon — Definition, Formula & Examples

The Gibbs phenomenon is the persistent overshoot (approximately 9%) that occurs when a Fourier series partial sum approximates a function at a jump discontinuity. No matter how many terms you add, the overshoot never disappears — it just moves closer to the discontinuity.

For a piecewise smooth function $f$ with a jump discontinuity, the $N$ th partial sum of its Fourier series overshoots the function value at the discontinuity by a factor that approaches $\frac{1}{\pi}\int_0^{\pi}\frac{\sin t}{t}\,dt \approx 1.0895$ (roughly 8.95% of the jump height) as $N \to \infty$ . The overshoot does not diminish with increasing $N$ ; instead, the spike narrows and concentrates nearer to the point of discontinuity.

Key Formula

\lim_{N\to\infty} \max_{x} S_N(x) \;=\; \frac{d}{2}\cdot\frac{2}{\pi}\int_0^{\pi}\frac{\sin t}{t}\,dt \;\approx\; 1.0895\cdot\frac{d}{2}

Where:

$S_N(x)$ = Partial sum of the Fourier series using the first N terms
$d$ = Total height of the jump discontinuity
$N$ = Number of Fourier terms in the partial sum

How It Works

Consider approximating a square wave with a finite Fourier series. Near each jump, the partial sum produces oscillations that peak above and dip below the true function value. As you increase the number of terms

N

, the peak overshoot stays at about 9% of the total jump, but the width of the spike shrinks proportionally to

1/N

. This means the

L^2

error decreases (the approximation converges in energy), yet the pointwise maximum error near the jump does not go to zero. The Gibbs phenomenon is not a flaw in the theory — the Fourier series still converges to

f

at every point of continuity and to the midpoint of the jump at the discontinuity itself.

Worked Example

Problem: A square wave alternates between

-1

and

1

with period

2\pi

. Estimate the maximum overshoot of its Fourier series partial sums near the jump at

x = 0

Step 1: Identify the jump height. The function goes from

-1

1

, so the total jump is

d = 2

d = 1 - (-1) = 2

Step 2: Apply the Gibbs overshoot factor. The partial sums overshoot to approximately

1.0895 \times \frac{d}{2}

above the midpoint of the jump.

\text{max value} \approx 1.0895 \times \frac{2}{2} = 1.0895

Step 3: Compute the overshoot relative to the true function value of

1

\text{overshoot} \approx 1.0895 - 1 = 0.0895 \approx 8.95\%

Answer: The Fourier partial sums overshoot the square wave by approximately 8.95% of the jump, reaching a peak value of about

1.09

instead of

1

Why It Matters

The Gibbs phenomenon matters in signal processing and image reconstruction, where sharp edges cause ringing artifacts. Engineers use window functions (Hann, Hamming) or sigma approximation specifically to suppress Gibbs overshoot when reconstructing signals from truncated Fourier data, such as in MRI imaging.

Common Mistakes

Mistake: Assuming the overshoot vanishes if you take enough Fourier terms.

Correction: The peak overshoot remains at ~9% of the jump regardless of how many terms you include. Only the width of the spike decreases. Convergence in

L^2

norm does not imply uniform convergence at discontinuities.