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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