Notebooks · posts/gibbs-overshoot

The Gibbs overshoot never shrinks

Adam Zettel · September 21, 2026 · written with an agent

Add more terms to the Fourier series of a square wave and the wiggle at the jump gets narrower, never shorter. It overshoots by 8.95% of the jump however many terms you take, and the number comes from an integral.

x max ∣ S_{m} (x) - f (x) ∣ \to \frac{2}{π} \int_{0}^{π} \frac{sin t}{t} d t - 1

Take the square wave that is $+ 1$ on $(0, π)$ and $- 1$ on $(- π, 0)$ . Its Fourier series is $\frac{4}{π} (sin x + \frac{1}{3} sin 3 x + \frac{1}{5} sin 5 x + \dots)$ : odd harmonics only, falling off like $1/ k$ . Sum the first five and you get a wobbly approximation. Sum eighty and the wobble has all but gone — except right next to the jump, where a spike of the same height as ever has moved in toward $x = 0$ and thinned.

Where the height comes from

The sum of the first $m$ odd harmonics has its first peak at exactly $x = π /2 m$ , where the derivative $\frac{2}{π} sin (2 m x) / sin x$ first vanishes. Evaluate the sum there and it is a Riemann sum, with $m$ panels, for $\frac{2}{π} \int_{0}^{π} \frac{s i n t}{t} d t = 1.17898$ . So the height of the first peak tends to a number that has nothing to do with $m$ : $(2/ π) Si (π)$ , against a wave of height 1. The overshoot is $0.179$ , which is $8.95%$ of the jump from $- 1$ to $1$ .

Why more terms cannot help

The peak sits at $π /2 m$ and is about that wide, so as $m$ grows it carries less and less area. In the mean-square sense the series converges: the integral of the squared error goes to zero. But at the peak itself the error stays $0.179$ for every $m$ . That is the difference between convergence in $L^{2}$ and uniform convergence, and a square wave, being discontinuous, has the first without the second. A smoothing filter (Fejér, Lanczos) trades the spike for a slower approach to the wave; nothing that keeps the partial sums as they are removes it.

What the program checks

The program checks three things. The peak of a thousand harmonics, summed directly, matches the integral computed by Simpson's rule with 2000 panels to a millionth. The peaks of ten and of a hundred harmonics are both above $1.17$ , so adding terms does not lower the peak. And the overshoot is $0.0895$ of the jump, to four places.

Look inside this program

Atlas
The program’s map: every function and which calls which.
There’s no recorded run of this program yet.

Shift+Enter in a cell runs it and moves to the next. Running a cell runs every cell above it too, as one program.

[1]lines 45–137

pi = 3.141592653589793

#| The first m odd harmonics of a unit square wave, at the points x.
def partial_sum(x, m) {
    s = 0.0 * x
    for j to m {
        k = 2.0 * j + 1.0
        s = s + sin(k * x) / k
    }
    return (4.0 / pi) * s
}

#| The same sum at a single point x.
def partial_sum_at(x, m) {
    s = 0.0
    for j to m {
        k = 2.0 * j + 1.0
        s = s + sin(k * x) / k
    }
    return (4.0 / pi) * s
}

#| sin(t) / t, with its limit at zero.
def sinc(t) {
    if t == 0.0 {
        return 1.0
    }
    return sin(t) / t
}

limit = (2.0 / pi) * take the integral of sinc(t) from 0 to pi with 2000 intervals
peak_10 = partial_sum_at(pi / 20.0, 10.0)
peak_100 = partial_sum_at(pi / 200.0, 100.0)
peak_1000 = partial_sum_at(pi / 2000.0, 1000.0)
overshoot = (limit - 1.0) / 2.0
# how far the peak rises above the wave's height of 1
overshoot_height = limit - 1.0
show limit
show peak_10
show peak_100
show peak_1000
show overshoot
show overshoot_height

# The explanation, as an animation: the corner of the wave just after the jump at
# x = 0, one partial sum morphing into the next. The peak moves toward the jump and
# narrows; compare its height with the dashed line.
# points across the corner
NX = 220
X = linspace(0.0, 1.0, NX)
HARMONICS = [2.0, 3.0, 5.0, 10.0, 20.0, 40.0, 80.0]

stage("Square wave at the jump, partial sums m = 5 to 80", 0.0, 1.0, 0.7, 1.25)
axes("x", "partial sum")
hline("wave", 1.0)
hline("limit", limit)
label("limit_name", "$\\tfrac{2}{\\pi}\\,\\mathrm{Si}(\\pi)$", 0.86, limit + 0.028)
curve("sum", X, partial_sum(X, 1.0))

fade_in("wave", 0.4)
say("the square wave is the dashed line at 1. One harmonic:")
draw("sum", 1.2)
wait(0.4)
for k to len(HARMONICS) {
    say("harmonics:", HARMONICS[k])
    morph("sum", X, partial_sum(X, HARMONICS[k]), 0.9)
    wait(0.35)
}
say("the peak has moved in and narrowed. Its height has not changed:")
together()
fade_in("limit", 0.6)
fade_in("limit_name", 0.6)
apart()
say("it sits at (2/pi) Si(pi) =", limit)
wait(1.2)

test "the peak of a thousand harmonics matches the integral (2/pi) Si(pi)" {
    assert_near(peak_1000, limit, 1.0e-6)
}
test "ten harmonics already overshoot by nearly as much: more terms do not help" {
    assert_gt(peak_10, 1.17)
    assert_gt(peak_100, 1.17)
}
test "the peak rises 0.179 above the wave" {
    # to the three digits the prose states
    assert_near(overshoot_height, 0.179, 0.0005)
}
test "the overshoot is 8.95% of the jump" {
    assert_near(overshoot, 0.0895, 1.0e-4)
}

# SPDX-License-Identifier: MIT OR Apache-2.0
# SPDX-FileCopyrightText: 2026 The Viv authors

pi = 3.141592653589793

#| The first m odd harmonics of a unit square wave, at the points x.
def partial_sum(x, m) {
    s = 0.0 * x
    for j to m {
        k = 2.0 * j + 1.0
        s = s + sin(k * x) / k
    }
    return (4.0 / pi) * s
}

#| The same sum at a single point x.
def partial_sum_at(x, m) {
    s = 0.0
    for j to m {
        k = 2.0 * j + 1.0
        s = s + sin(k * x) / k
    }
    return (4.0 / pi) * s
}

#| sin(t) / t, with its limit at zero.
def sinc(t) {
    if t == 0.0 {
        return 1.0
    }
    return sin(t) / t
}

limit = (2.0 / pi) * take the integral of sinc(t) from 0 to pi with 2000 intervals
peak_10 = partial_sum_at(pi / 20.0, 10.0)
peak_100 = partial_sum_at(pi / 200.0, 100.0)
peak_1000 = partial_sum_at(pi / 2000.0, 1000.0)
overshoot = (limit - 1.0) / 2.0
# how far the peak rises above the wave's height of 1
overshoot_height = limit - 1.0
show limit
show peak_10
show peak_100
show peak_1000
show overshoot
show overshoot_height

# The explanation, as an animation: the corner of the wave just after the jump at
# x = 0, one partial sum morphing into the next. The peak moves toward the jump and
# narrows; compare its height with the dashed line.
# points across the corner
NX = 220
X = linspace(0.0, 1.0, NX)
HARMONICS = [2.0, 3.0, 5.0, 10.0, 20.0, 40.0, 80.0]

stage("Square wave at the jump, partial sums m = 5 to 80", 0.0, 1.0, 0.7, 1.25)
axes("x", "partial sum")
hline("wave", 1.0)
hline("limit", limit)
label("limit_name", "$\\tfrac{2}{\\pi}\\,\\mathrm{Si}(\\pi)$", 0.86, limit + 0.028)
curve("sum", X, partial_sum(X, 1.0))

fade_in("wave", 0.4)
say("the square wave is the dashed line at 1. One harmonic:")
draw("sum", 1.2)
wait(0.4)
for k to len(HARMONICS) {
    say("harmonics:", HARMONICS[k])
    morph("sum", X, partial_sum(X, HARMONICS[k]), 0.9)
    wait(0.35)
}
say("the peak has moved in and narrowed. Its height has not changed:")
together()
fade_in("limit", 0.6)
fade_in("limit_name", 0.6)
apart()
say("it sits at (2/pi) Si(pi) =", limit)
wait(1.2)

test "the peak of a thousand harmonics matches the integral (2/pi) Si(pi)" {
    assert_near(peak_1000, limit, 1.0e-6)
}
test "ten harmonics already overshoot by nearly as much: more terms do not help" {
    assert_gt(peak_10, 1.17)
    assert_gt(peak_100, 1.17)
}
test "the peak rises 0.179 above the wave" {
    # to the three digits the prose states
    assert_near(overshoot_height, 0.179, 0.0005)
}
test "the overshoot is 8.95% of the jump" {
    assert_near(overshoot, 0.0895, 1.0e-4)
}

# SPDX-License-Identifier: MIT OR Apache-2.0
# SPDX-FileCopyrightText: 2026 The Viv authors