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The Gibbs’ Phenomenon
Eli Dean & Britton Girard
Math 572, Fall 2013
December 4, 2013
1. Introduction 1
SNf(x) :=∑|n|≤N
f(n)einx :=∑|n|≤N
1
2π
π∫−π
f(x)e−inx dx
einx
denotes the partial Fourier sum for f at the point x
Can use Fourier Sums to approximate f : T→ R under theappropriate conditions
1. Introduction 2
Defn: Pointwise Convergence of Functions:
A series of functions {fn}n∈N converges pointwise to f on the set Dif ∀ε > 0 and ∀x ∈ D, ∃N ∈ N such that |fn(x)− f(x)| < ε∀n ≥ N .
Defn: Uniform Convergence of Functions:
A series of functions {fn}n∈N converges uniformly to f on the set Dif ∀ε > 0, ∃N ∈ N such that |fn(x)− f(x)| < ε∀n ≥ N and ∀x ∈ D.
−4 −3 −2 −1 0 1 2 3 4−1.5
−1
−0.5
0
0.5
1
1.5
Figure : Pointwise Conv.
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
−0.2
0
0.2
0.4
0.6
0.8
Figure : Uniform Conv.
1. Introduction 3
Some movies to illustrate the difference. (First Uniform thenPointwise)
Persistent “bump” in second illustrates Gibbs’ Phenomenon.
1. Introduction 3
Some movies to illustrate the difference. (First Uniform thenPointwise)
Persistent “bump” in second illustrates Gibbs’ Phenomenon.
2. Gibbs’ Phenomenon: A Brief History 1
Basic Background
1848: Property of overshooting discovered by Wilbraham
1899: Gibbs brings attention to behavior of Fourier Series (Gibbsobserved same behavior as Wilbraham but by studying a differentfunction)
1906: Maxime Brocher shows that the phenomenon occurs forgeneral Fourier Series around a jump discontinuity
2. Gibbs’ Phenomenon: A Brief History 2
Key Players and Contributions
Lord Kelvin: Constructed two machines while studying tide heights asa function of time, h(t).
Machine capable of computing periodic function h(t) using FourierCoefficientsConstructed a “Harmonic Analyzer” capable of computing Fouriercoefficients of past tide height functions, h(t).
A. A. Michelson:
Elaborated on Kelvin’s device and constructed a machine “which wouldsave considerable time and labour involved in calculations. . . of theresultant of a large number of simple harmonic motions.”Using first 80 coefficients of sawtooth wave, Michelson’s machineclosely approximated the sawtooth function except for two blips nearthe points of discontinuity
2. Gibbs’ Phenomenon: A Brief History 3
Key Players and Contributions Cont.
Wilbraham
1848: Wilbraham investigated the equation:
y = cos(x) +cos(3x)
3+
cos(5x)
5+ ...+
cos((2n− 1)x)
2n− 1+ ...
Discovered that “at a distance of π4 alternately above and below it,
joined by perpendiculars which are themselves part of the locus” the
equation overshoots π4 by a distance of: 1
2
∣∣∣∫∞π sin(x)x dx
∣∣∣Also suggested that similar analysis of
y = 2(sin(x)− sin(2x)
2 + sin(3x)3 − ...+ (−1)n+1 sin(nx)
n
)(An equation
explored later on) would lead to an analogous resultThese discoveries gained little attention at the time
2. Gibbs’ Phenomenon: A Brief History 4
Key Players and Contributions Cont.
J. Willard Gibbs
1898: Published and article in Nature investigating the behavior of thefunction given by:
y = 2
(sin(x)− sin(2x)
2+
sin(3x)
3− ...+ (−1)n+1 sin(nx)
n
)Gibbs observed in this first article that the limiting behavior of thefunction (sawtooth) had “vertical portions, which are bisected the axisof X, [extending] beyond the points where they meet the inclinedportions, their total lengths being express by four times the definite
integral
π∫0
sin(u)
udu.”
2. Gibbs’ Phenomenon: A Brief History 5
Key Players and Contributions Cont.
Brocher
1906: In an article in Annals of Mathematics, Brocher demonstratedthat Gibbs’ Phenomenon will be observed in any Fourier Series of afunction f with a jump discontinuity saying that the limiting curve ofthe approximating curves has a vertical line that “has to be producedbeyond these points by an amount that bears a devinite ratio to themagnitude of the jump.”Before Brocher, Gibbs’ Phenomenon had only been observed in specificseries without any generalization of the concept
3. Gibbs’ Phenomenon: An Example 1
Consider the square wave function given by:
g(x) :=
{−1 if − π ≤ x < 01 if 0 ≤ x < π
Goal: Analytically prove that Gibbs’ Phenomenon occurs at jumpdiscontinuities of this 2π-periodic function
3. Gibbs’ Phenomenon: An Example 2
Using the definition an := 12π
π∫−π
f(x)e−inx dx, basic Calculus yields
SNf(x) =
N∑n=−N
aneinθ =
1
π
∑|n|≤Nn is odd
(2
n
)einθ
i
=4
π
M∑m=1
sin((2m− 1)θ)
2m− 1
where M is the largest integer such that 2M − 1 ≤ N .
3. Gibbs’ Phenomenon: An Example 3
Here we notice that
sin((2m− 1)θ)
2m− 1=
∫ x
0cos((2m− 1)θ) dθ
which yields the following identity:
S2M−1g(x) =4
π
M∑m=1
sin((2m− 1)x)
2m− 1
=4
π
M∑m=1
∫ x
0cos((2m− 1)θ) dθ
=4
π
∫ x
0
M∑m=1
cos((2m− 1)θ) dθ
3. Gibbs’ Phenomenon: An Example 4
Now using the identity
M∑m=1
cos((2m− 1)x) =sin(2Mx)
2 sin(x)
we get the following:
S2M−1g(x) =4
π
∫ x
0
M∑m=1
cos((2m− 1)θ) dθ
=4
π
∫ x
0
sin(2Mθ)
2 sin(θ)dθ
3. Gibbs’ Phenomenon: An Example 5
At this point, using basic analysis of the derivative of S2M−1g(x), we seethat there are local maximum and minimum values at
xM,+ =π
2Mand xM,− = − π
2MHere, using the substitution u = 2Mθ and the fact that for u ≈ 0,sin(u) ≈ u, we get that for large M
S2M−1g(xm,+) ≈2
π
∫ π
0
sin(θ)
θdθ
and
S2M−1g(xm,−) ≈ −2
π
∫ π
0
sin(θ)
θdθ
Further, the value 2π
π∫0
sin(θ)
θdθ ≈ 1.17898. Have shown that
∀M >> 1, M ∈ N, ∃x = xM,+ such that S2M−1g(xm,+) ≈ 1.17898.⇒ is always a “blip” near the jump discontinuity, namely at x = π
2M ,where the value S2M−1g(x) is not near 1!
3. Gibbs’ Phenomenon: An Example 6
Have shown analytically that a “blip” persists for all S2M−1g near thejump discontinuity.
Demonstrated again by square wave
We can also see the same behavior in the 2π-periodic sawtoothfunction given by f(x) = x on the interval [−π, π).
3. Gibbs’ Phenomenon: An Example 6
Have shown analytically that a “blip” persists for all S2M−1g near thejump discontinuity.
Demonstrated again by square wave
We can also see the same behavior in the 2π-periodic sawtoothfunction given by f(x) = x on the interval [−π, π).
3. Gibbs’ Phenomenon: An Example 7
For both of the square wave and sawtooth functions, we have shown thata “blip” of constant size persists
⇒ while SMg → g pointwise, we now know that SMg 6→ g uniformly sincefor small enough neighborhoods, the “blip” is always outside theboundaries. (revisit square wave animation)
4. Gibbs’ Phenomenon: General Principles 1
Kernels And ConvolutionsKernels:
Defn: (Good Kernel) A family {Kn}n∈N real-valued integrablefunctions on the circle is a family of good kernels if the following hold:
(i) ∀n ∈ N, 12π
π∫−π
Kn(θ) dθ = 1.
(ii) ∃M > 0 such that ∀n ∈ N,π∫−π
|Kn(θ)| dθ ≤M .
(iii) ∀δ > 0,
∫δ≤|θ|<π
|Kn(θ)| dθ → 0 as n→∞.
Dirichlet: DN (θ) =∑|n|≤N
einθ
⇒π∫−π
DN (θ) dθ =∑|n|≤N
π∫−π
einθ dθ
Fejer: FN (θ) = [D0(θ) +D1(θ) + ...+DN−1(θ)]/NRecall: Dirichlet Kernel is NOT a good kernel and Fejer Kernel is.
4. Gibbs’ Phenomenon: General Principles 1
Kernels And ConvolutionsKernels:
Defn: (Good Kernel) A family {Kn}n∈N real-valued integrablefunctions on the circle is a family of good kernels if the following hold:
(i) ∀n ∈ N, 12π
π∫−π
Kn(θ) dθ = 1.
(ii) ∃M > 0 such that ∀n ∈ N,π∫−π
|Kn(θ)| dθ ≤M .
(iii) ∀δ > 0,
∫δ≤|θ|<π
|Kn(θ)| dθ → 0 as n→∞.
Dirichlet: DN (θ) =∑|n|≤N
einθ
⇒π∫−π
DN (θ) dθ =∑|n|≤N
π∫−π
einθ dθ
Fejer: FN (θ) = [D0(θ) +D1(θ) + ...+DN−1(θ)]/NRecall: Dirichlet Kernel is NOT a good kernel and Fejer Kernel is.
4. Gibbs’ Phenomenon: General Principles 1
Kernels And ConvolutionsKernels:
Defn: (Good Kernel) A family {Kn}n∈N real-valued integrablefunctions on the circle is a family of good kernels if the following hold:
(i) ∀n ∈ N, 12π
π∫−π
Kn(θ) dθ = 1.
(ii) ∃M > 0 such that ∀n ∈ N,π∫−π
|Kn(θ)| dθ ≤M .
(iii) ∀δ > 0,
∫δ≤|θ|<π
|Kn(θ)| dθ → 0 as n→∞.
Dirichlet: DN (θ) =∑|n|≤N
einθ
⇒π∫−π
DN (θ) dθ =∑|n|≤N
π∫−π
einθ dθ
Fejer: FN (θ) = [D0(θ) +D1(θ) + ...+DN−1(θ)]/NRecall: Dirichlet Kernel is NOT a good kernel and Fejer Kernel is.
4. Gibbs’ Phenomenon: General Principles 1
Kernels And ConvolutionsKernels:
Defn: (Good Kernel) A family {Kn}n∈N real-valued integrablefunctions on the circle is a family of good kernels if the following hold:
(i) ∀n ∈ N, 12π
π∫−π
Kn(θ) dθ = 1.
(ii) ∃M > 0 such that ∀n ∈ N,π∫−π
|Kn(θ)| dθ ≤M .
(iii) ∀δ > 0,
∫δ≤|θ|<π
|Kn(θ)| dθ → 0 as n→∞.
Dirichlet: DN (θ) =∑|n|≤N
einθ
⇒π∫−π
DN (θ) dθ =∑|n|≤N
π∫−π
einθ dθ
Fejer: FN (θ) = [D0(θ) +D1(θ) + ...+DN−1(θ)]/N
Recall: Dirichlet Kernel is NOT a good kernel and Fejer Kernel is.
4. Gibbs’ Phenomenon: General Principles 1
Kernels And ConvolutionsKernels:
Defn: (Good Kernel) A family {Kn}n∈N real-valued integrablefunctions on the circle is a family of good kernels if the following hold:
(i) ∀n ∈ N, 12π
π∫−π
Kn(θ) dθ = 1.
(ii) ∃M > 0 such that ∀n ∈ N,π∫−π
|Kn(θ)| dθ ≤M .
(iii) ∀δ > 0,
∫δ≤|θ|<π
|Kn(θ)| dθ → 0 as n→∞.
Dirichlet: DN (θ) =∑|n|≤N
einθ
⇒π∫−π
DN (θ) dθ =∑|n|≤N
π∫−π
einθ dθ
Fejer: FN (θ) = [D0(θ) +D1(θ) + ...+DN−1(θ)]/NRecall: Dirichlet Kernel is NOT a good kernel and Fejer Kernel is.
4. Gibbs’ Phenomenon: General Principles 2
Kernels And Convolutions Cont.Convolution:
Defn: The convolution of two functions, f and g is given by:
(f ∗ g)(θ) = 1
2π
π∫−π
f(y)g(θ − y) dy (1)
⇒ SMf = (DM ∗ f)(θ)
Defn: σMf(θ) := [S0f(θ) + S1f(θ) + ...+ SM−1f(θ)]/M
From a previous homework, we also know that FM ∗ f = σMf⇒ FM ∗ f is of sort an averaging of the Dirichlet kernels, or moreimportantly, the partial Fourier sums.
4. Gibbs’ Phenomenon: General Principles 2
Kernels And Convolutions Cont.Convolution:
Defn: The convolution of two functions, f and g is given by:
(f ∗ g)(θ) = 1
2π
π∫−π
f(y)g(θ − y) dy (1)
⇒ SMf = (DM ∗ f)(θ)
Defn: σMf(θ) := [S0f(θ) + S1f(θ) + ...+ SM−1f(θ)]/M
From a previous homework, we also know that FM ∗ f = σMf⇒ FM ∗ f is of sort an averaging of the Dirichlet kernels, or moreimportantly, the partial Fourier sums.
4. Gibbs’ Phenomenon: General Principles 2
Kernels And Convolutions Cont.Convolution:
Defn: The convolution of two functions, f and g is given by:
(f ∗ g)(θ) = 1
2π
π∫−π
f(y)g(θ − y) dy (1)
⇒ SMf = (DM ∗ f)(θ)
Defn: σMf(θ) := [S0f(θ) + S1f(θ) + ...+ SM−1f(θ)]/M
From a previous homework, we also know that FM ∗ f = σMf⇒ FM ∗ f is of sort an averaging of the Dirichlet kernels, or moreimportantly, the partial Fourier sums.
4. Gibbs’ Phenomenon: General Principles 2
Kernels And Convolutions Cont.Convolution:
Defn: The convolution of two functions, f and g is given by:
(f ∗ g)(θ) = 1
2π
π∫−π
f(y)g(θ − y) dy (1)
⇒ SMf = (DM ∗ f)(θ)
Defn: σMf(θ) := [S0f(θ) + S1f(θ) + ...+ SM−1f(θ)]/M
From a previous homework, we also know that FM ∗ f = σMf
⇒ FM ∗ f is of sort an averaging of the Dirichlet kernels, or moreimportantly, the partial Fourier sums.
4. Gibbs’ Phenomenon: General Principles 2
Kernels And Convolutions Cont.Convolution:
Defn: The convolution of two functions, f and g is given by:
(f ∗ g)(θ) = 1
2π
π∫−π
f(y)g(θ − y) dy (1)
⇒ SMf = (DM ∗ f)(θ)
Defn: σMf(θ) := [S0f(θ) + S1f(θ) + ...+ SM−1f(θ)]/M
From a previous homework, we also know that FM ∗ f = σMf⇒ FM ∗ f is of sort an averaging of the Dirichlet kernels, or moreimportantly, the partial Fourier sums.
4. Gibbs’ Phenomenon: General Principles 3
The Problem is Quite GeneralBrocher showed Gibbs’ Phenomenon occurs with any 2π-periodicpiecewise-continuous function with a jump discontinuity
Overshoot is proportional to height of jump:
Square Wave: Overshoot ≈ 0.17898⇒ 0.17898 = k(2)⇒ k = 0.08949⇒ Overshoot/Undershoot given by ≈ 0.08949(f(x+d )− f(x
−d ) where
xd is point of discontinuitySuggests an overshoot of ≈ 0.08949(2π) ≈ 0.56228 for a maximumheight of ≈ 3.70228
Thus, need a way to resolve this for modeling of periodic phenomenato be practical
Using Fejer Kernel averages Dirichlet kernels and could therefore be away of eliminating this phenomenon
animation experiment
5. Conclusion 1
Gibbs’ Phenomenon observed in all 2π-periodic functions with a jumpdiscontinuity
In some cases, it is quite basic to show it’s persistence analytically
Causes nastiness in approximations of functions by Fourier series, butis easily remedied with averaging using Fejer Kernel