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Page 1: The Gibbs' Phenomenoncrisp/courses/wavelets/fall13/...2. Gibbs’ Phenomenon: A Brief History 2 Key Players and Contributions Lord Kelvin: Constructed two machines while studying tide

The Gibbs’ Phenomenon

Eli Dean & Britton Girard

Math 572, Fall 2013

December 4, 2013

Page 2: The Gibbs' Phenomenoncrisp/courses/wavelets/fall13/...2. Gibbs’ Phenomenon: A Brief History 2 Key Players and Contributions Lord Kelvin: Constructed two machines while studying tide

1. Introduction 1

SNf(x) :=∑|n|≤N

f(n)einx :=∑|n|≤N

1

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

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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.

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1. Introduction 3

Some movies to illustrate the difference. (First Uniform thenPointwise)

Persistent “bump” in second illustrates Gibbs’ Phenomenon.

Page 5: The Gibbs' Phenomenoncrisp/courses/wavelets/fall13/...2. Gibbs’ Phenomenon: A Brief History 2 Key Players and Contributions Lord Kelvin: Constructed two machines while studying tide

1. Introduction 3

Some movies to illustrate the difference. (First Uniform thenPointwise)

Persistent “bump” in second illustrates Gibbs’ Phenomenon.

Page 6: The Gibbs' Phenomenoncrisp/courses/wavelets/fall13/...2. Gibbs’ Phenomenon: A Brief History 2 Key Players and Contributions Lord Kelvin: Constructed two machines while studying tide

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

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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

Page 8: The Gibbs' Phenomenoncrisp/courses/wavelets/fall13/...2. Gibbs’ Phenomenon: A Brief History 2 Key Players and Contributions Lord Kelvin: Constructed two machines while studying tide

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

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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.”

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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

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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

Page 12: The Gibbs' Phenomenoncrisp/courses/wavelets/fall13/...2. Gibbs’ Phenomenon: A Brief History 2 Key Players and Contributions Lord Kelvin: Constructed two machines while studying tide

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 .

Page 13: The Gibbs' Phenomenoncrisp/courses/wavelets/fall13/...2. Gibbs’ Phenomenon: A Brief History 2 Key Players and Contributions Lord Kelvin: Constructed two machines while studying tide

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

Page 14: The Gibbs' Phenomenoncrisp/courses/wavelets/fall13/...2. Gibbs’ Phenomenon: A Brief History 2 Key Players and Contributions Lord Kelvin: Constructed two machines while studying tide

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

Page 15: The Gibbs' Phenomenoncrisp/courses/wavelets/fall13/...2. Gibbs’ Phenomenon: A Brief History 2 Key Players and Contributions Lord Kelvin: Constructed two machines while studying tide

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!

Page 16: The Gibbs' Phenomenoncrisp/courses/wavelets/fall13/...2. Gibbs’ Phenomenon: A Brief History 2 Key Players and Contributions Lord Kelvin: Constructed two machines while studying tide

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).

Page 17: The Gibbs' Phenomenoncrisp/courses/wavelets/fall13/...2. Gibbs’ Phenomenon: A Brief History 2 Key Players and Contributions Lord Kelvin: Constructed two machines while studying tide

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).

Page 18: The Gibbs' Phenomenoncrisp/courses/wavelets/fall13/...2. Gibbs’ Phenomenon: A Brief History 2 Key Players and Contributions Lord Kelvin: Constructed two machines while studying tide

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)

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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.

Page 20: The Gibbs' Phenomenoncrisp/courses/wavelets/fall13/...2. Gibbs’ Phenomenon: A Brief History 2 Key Players and Contributions Lord Kelvin: Constructed two machines while studying tide

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.

Page 21: The Gibbs' Phenomenoncrisp/courses/wavelets/fall13/...2. Gibbs’ Phenomenon: A Brief History 2 Key Players and Contributions Lord Kelvin: Constructed two machines while studying tide

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.

Page 22: The Gibbs' Phenomenoncrisp/courses/wavelets/fall13/...2. Gibbs’ Phenomenon: A Brief History 2 Key Players and Contributions Lord Kelvin: Constructed two machines while studying tide

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.

Page 23: The Gibbs' Phenomenoncrisp/courses/wavelets/fall13/...2. Gibbs’ Phenomenon: A Brief History 2 Key Players and Contributions Lord Kelvin: Constructed two machines while studying tide

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.

Page 24: The Gibbs' Phenomenoncrisp/courses/wavelets/fall13/...2. Gibbs’ Phenomenon: A Brief History 2 Key Players and Contributions Lord Kelvin: Constructed two machines while studying tide

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

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.

Page 25: The Gibbs' Phenomenoncrisp/courses/wavelets/fall13/...2. Gibbs’ Phenomenon: A Brief History 2 Key Players and Contributions Lord Kelvin: Constructed two machines while studying tide

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

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.

Page 26: The Gibbs' Phenomenoncrisp/courses/wavelets/fall13/...2. Gibbs’ Phenomenon: A Brief History 2 Key Players and Contributions Lord Kelvin: Constructed two machines while studying tide

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

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.

Page 27: The Gibbs' Phenomenoncrisp/courses/wavelets/fall13/...2. Gibbs’ Phenomenon: A Brief History 2 Key Players and Contributions Lord Kelvin: Constructed two machines while studying tide

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

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.

Page 28: The Gibbs' Phenomenoncrisp/courses/wavelets/fall13/...2. Gibbs’ Phenomenon: A Brief History 2 Key Players and Contributions Lord Kelvin: Constructed two machines while studying tide

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

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.

Page 29: The Gibbs' Phenomenoncrisp/courses/wavelets/fall13/...2. Gibbs’ Phenomenon: A Brief History 2 Key Players and Contributions Lord Kelvin: Constructed two machines while studying tide

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

Page 30: The Gibbs' Phenomenoncrisp/courses/wavelets/fall13/...2. Gibbs’ Phenomenon: A Brief History 2 Key Players and Contributions Lord Kelvin: Constructed two machines while studying tide

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