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### Transcript of The Gibbs' Phenomenon crisp/courses/wavelets/fall13/... 2. Gibbs’ Phenomenon: A Brief History...

• The Gibbs’ Phenomenon

Eli Dean & Britton Girard

Math 572, Fall 2013

December 4, 2013

• 1. Introduction 1

SNf(x) := ∑ |n|≤N

f̂(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 the appropriate conditions

• 1. Introduction 2

Defn: Pointwise Convergence of Functions:

A series of functions {fn}n∈N converges pointwise to f on the set D if ∀� > 0 and ∀x ∈ D, ∃N ∈ N such that |fn(x)− f(x)| < �∀n ≥ N .

Defn: Uniform Convergence of Functions:

A series of functions {fn}n∈N converges uniformly to f on the set D if ∀� > 0, ∃N ∈ N such that |fn(x)− f(x)| < �∀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 then Pointwise)

Persistent “bump” in second illustrates Gibbs’ Phenomenon.

• 1. Introduction 3

Some movies to illustrate the difference. (First Uniform then Pointwise)

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 (Gibbs observed same behavior as Wilbraham but by studying a different function)

1906: Maxime Brocher shows that the phenomenon occurs for general 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 as a function of time, h(t).

Machine capable of computing periodic function h(t) using Fourier Coefficients Constructed a “Harmonic Analyzer” capable of computing Fourier coefficients of past tide height functions, h(t).

A. A. Michelson:

Elaborated on Kelvin’s device and constructed a machine “which would save considerable time and labour involved in calculations. . . of the resultant of a large number of simple harmonic motions.” Using first 80 coefficients of sawtooth wave, Michelson’s machine closely approximated the sawtooth function except for two blips near the 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 result These 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 the function 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 the function (sawtooth) had “vertical portions, which are bisected the axis of X, [extending] beyond the points where they meet the inclined portions, their total lengths being express by four times the definite

integral

π∫ 0

sin(u)

u du.”

• 2. Gibbs’ Phenomenon: A Brief History 5

Key Players and Contributions Cont.

Brocher

1906: In an article in Annals of Mathematics, Brocher demonstrated that Gibbs’ Phenomenon will be observed in any Fourier Series of a function f with a jump discontinuity saying that the limiting curve of the approximating curves has a vertical line that “has to be produced beyond these points by an amount that bears a devinite ratio to the magnitude of the jump.” Before Brocher, Gibbs’ Phenomenon had only been observed in specific series 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 < 0 1 if 0 ≤ x < π

Goal: Analytically prove that Gibbs’ Phenomenon occurs at jump discontinuities of this 2π-periodic function

• 3. Gibbs’ Phenomenon: An Example 2

Using the definition an := 1 2π

π∫ −π

f(x)e−inx dx, basic Calculus yields

SNf(x) =

N∑ n=−N

ane inθ =

1

π

∑ |n|≤N n 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 0

cos((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 0

cos((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 see that there are local maximum and minimum values at

xM,+ = π

2M and xM,− = −

π

2M Here, 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)

θ dθ

and

S2M−1g(xm,−) ≈ − 2

π

∫ π 0

sin(θ)

θ dθ

Further, the value 2π

π∫ 0

sin(θ)

θ dθ ≈ 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 = π2M , 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 the jump discontinuity.

Demonstrated again by square wave

We can also see the same behavior in the 2π-periodic sawtooth function 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 the jump discontinuity.

Demonstrated again by square wave

We can also see the same behavior in the 2π-periodic sawtooth function 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 that a “blip” of constant size persists

⇒ while SMg → g pointwise, we now know that SMg 6→ g uniformly since for small enough neighborhoods, the “blip” is always outside the boundaries. (revisit square wave animation)

• 4. Gibbs’ Phenomenon: General Principles 1

Kernels And Convolutions Kernels:

Defn: (Good Kernel) A family {Kn}n∈N real-valued integrable functions on the circle is a family of good kernels if the following hold:

(i) ∀n ∈ N, 1 2π

π∫ −π

Kn(θ) dθ = 1.

(ii) ∃M > 0 such that ∀n ∈ N, π∫ −π

|Kn(θ)| dθ ≤M .

(iii) ∀δ > 0, ∫

δ≤|θ|

• 4. Gibbs’ Phenomenon: General Principles 1

Kernels And Convolutions Kernels:

Defn: (Good Kernel) A family {Kn}n∈N real-valued integrable functions on the circle is a family of good kernels if the following hold:

(i) ∀n ∈ N, 1 2π

π∫ −π

Kn(θ) dθ = 1.

(ii) ∃M > 0 such that ∀n ∈ N, π∫ −π

|Kn(θ)| dθ ≤M .

(iii) ∀δ > 0, ∫

δ≤|θ|

• 4. Gibbs’ Phenomenon: General Principles 1

Kernels And Convolutions Kernels:

Defn: (Good Kernel) A family {Kn}n∈N real-valued integrable functions on the circle is a family of good kernels if the following hold:

(i) ∀n ∈ N, 1 2π

π∫ −π

Kn(θ) dθ = 1.

(ii) ∃M > 0 such that ∀n ∈ N, π∫ −π

|Kn(θ)| dθ ≤M .

(iii) ∀δ > 0, ∫

δ≤|θ|

• 4. Gibbs