Download - The Gibbs' Phenomenon - University of New Mexicocrisp/courses/wavelets/fall13... · 2. Gibbs’ Phenomenon: A Brief History 2 Key Players and Contributions Lord Kelvin: Constructed

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.

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


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


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



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.”



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


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 .


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


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


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!


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).


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.




(i) ∀n ∈ N, 12π

π∫−π

Kn(θ) dθ = 1.


|Kn(θ)| dθ ≤M .

(iii) ∀δ > 0,

∫δ≤|θ|<π

|Kn(θ)| dθ → 0 as n→∞.


einθ

⇒π∫−π


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.




(i) ∀n ∈ N, 12π

π∫−π

Kn(θ) dθ = 1.


|Kn(θ)| dθ ≤M .

(iii) ∀δ > 0,

∫δ≤|θ|<π

|Kn(θ)| dθ → 0 as n→∞.


einθ

⇒π∫−π


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.


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.




(f ∗ g)(θ) = 1

2π

π∫−π

f(y)g(θ − y) dy (1)

⇒ SMf = (DM ∗ f)(θ)


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.




(f ∗ g)(θ) = 1

2π

π∫−π

f(y)g(θ − y) dy (1)

⇒ SMf = (DM ∗ f)(θ)


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.


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

Download - The Gibbs' Phenomenon - University of New Mexicocrisp/courses/wavelets/fall13... · 2. Gibbs’ Phenomenon: A Brief History 2 Key Players and Contributions Lord Kelvin: Constructed

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