Fourier Transform 20 | Gibbs Phenomenon

Gibbs phenomenon is the stubborn, built-in overshoot that appears when a Fourier series approximates a function with a jump discontinuity—and it does not disappear no matter how many Fourier terms are used. Away from the jump, increasing the number of terms improves the approximation pointwise, but near the discontinuity the partial sums keep oscillating with a characteristic “ringing” pattern. The overshoot shifts toward the jump as the number of terms grows, yet its size stays essentially the same in the limit, so the approximation never becomes uniformly accurate across the jump.

To make that concrete, the discussion focuses on a standard step function on the interval  to , extended periodically. The function takes value 1 on one side of the origin and 0 on the other, creating a negative jump at x = 0. Fourier series theory already guarantees that at the jump point itself the series converges to the midpoint value (here, 1/2). The surprising part is what happens just next to the jump: even when the series is computed with very large truncation n (the example contrasts n = 40 with n = 100), the overshoot remains at roughly the same magnitude.

The core of the proof strategy is to estimate the truncated Fourier series near the jump using a closed-form expression for this particular step function. Because the function is essentially an odd function up to a vertical shift, the Fourier expansion involves only sine terms with odd frequencies. With an odd truncation n, the partial sum can be written as a finite series of the form  sin(kx)/k over odd k. The overshoot near the jump is then tied to estimating this oscillatory sum.

The argument turns the sum into an integral estimate. By introducing an integral variable and rewriting trigonometric terms using Euler’s formula, the analysis reduces the complicated oscillatory sum to a ratio involving sine functions. This leads to an expression whose behavior is governed by the sinc function and, after integration, by the sine integral (denoted SI). The sine integral has a pronounced maximum at , and that maximum controls where the Fourier partial sum achieves its strongest deviation.

As a result, the most significant overshoot occurs at a point scaled with n: x = /(n+1). Evaluating the Fourier partial sum at that location yields a quantitative bound on the overshoot size. For the unit jump from 1 down to 0, the calculation produces a deficit of about 0.089 from the midpoint value, meaning the partial sums undershoot/overshoot by roughly 8.9% relative to the jump height. Crucially, this percentage is effectively independent of n: increasing n only moves the location of the ringing closer to the discontinuity. In the limit as n  , the overshoot’s position collapses onto the jump, so the effect vanishes pointwise everywhere except at the discontinuity—yet the characteristic Gibbs magnitude remains a mathematical constant of the approximation process.