The Gibbs Phenomenon

Transcription

1 The Gibbs Phenomenon Eli Dean & Britton Girard Math 572, Fall 2013 December 4, 2013

2 1. Introduction 1 S N f(x) := f(n)e inx := 1 2π n N n N denotes the partial Fourier sum for f at the point x f(x)e inx dx e inx Can use Fourier Sums to approximate f : T R under the appropriate conditions

3 1. Introduction 2 Def n : Pointwise Convergence of Functions: A series of functions {f n } n N converges pointwise to f on the set D if ɛ > 0 and x D, N N such that f n (x) f(x) < ɛ n N. Def n : Uniform Convergence of Functions: A series of functions {f n } n N converges uniformly to f on the set D if ɛ > 0, N N such that f n (x) f(x) < ɛ n N and x D Figure : Pointwise Conv Figure : Uniform Conv.

4 1. Introduction 3 Some movies to illustrate the difference. (First Uniform then Pointwise)

5 1. Introduction 3 Some movies to illustrate the difference. (First Uniform then Pointwise) Persistent bump in second illustrates Gibbs Phenomenon.

6 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

7 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

8 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) cos((2n 1)x) 2n Discovered that at a distance of π 4 alternately above and below it, joined by perpendiculars which are themselves part of the locus the equation overshoots π 4 by a distance of: 1 2 sin(x) π x dx Also suggested ( that similar analysis of ) y = 2 sin(x) sin(2x) 2 + sin(3x) ( 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

9 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) + sin(3x) ) n+1 sin(nx)... + ( 1) 2 3 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 0 sin(u) u du.

10 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

11 3. Gibbs Phenomenon: An Example 1 Consider the square wave function given by: { 1 if π x < 0 g(x) := 1 if 0 x < π Goal: Analytically prove that Gibbs Phenomenon occurs at jump discontinuities of this 2π-periodic function

12 3. Gibbs Phenomenon: An Example 2 Using the definition a n := 1 2π S N f(x) = = 4 π N n= N M m=1 f(x)e inx dx, basic Calculus yields a n e inθ = 1 π n N n is odd sin((2m 1)θ) 2m 1 ( ) 2 e inθ n i where M is the largest integer such that 2M 1 N.

13 3. Gibbs Phenomenon: An Example 3 Here we notice that sin((2m 1)θ) 2m 1 which yields the following identity: S 2M 1 g(x) = 4 π = 4 π = 4 π = M x 0 m=1 x M m=1 x 0 cos((2m 1)θ) dθ sin((2m 1)x) 2m 1 0 cos((2m 1)θ) dθ M cos((2m 1)θ) dθ m=1

14 3. Gibbs Phenomenon: An Example 4 Now using the identity we get the following: M m=1 S 2M 1 g(x) = 4 π cos((2m 1)x) = sin(2mx) 2 sin(x) = 4 π x 0 m=1 x 0 M cos((2m 1)θ) dθ sin(2m θ) 2 sin(θ) dθ

15 3. Gibbs Phenomenon: An Example 5 At this point, using basic analysis of the derivative of S 2M 1 g(x), we see that there are local maximum and minimum values at x M,+ = π and x M, = π 2M 2M Here, using the substitution u = 2Mθ and the fact that for u 0, sin(u) u, we get that for large M Further, the value 2 π S 2M 1 g(x m,+ ) 2 π and S 2M 1 g(x m, ) 2 π 0 sin(θ) θ 0 0 sin(θ) θ dθ sin(θ) θ dθ dθ Have shown that M >> 1, M N, x = x M,+ such that S 2M 1 g(x m,+ ) is always a blip near the jump discontinuity, namely at x = where the value S 2M 1 g(x) is not near 1! π 2M,

16 3. Gibbs Phenomenon: An Example 6 Have shown analytically that a blip persists for all S 2M 1 g near the jump discontinuity. Demonstrated again by square wave

17 3. Gibbs Phenomenon: An Example 6 Have shown analytically that a blip persists for all S 2M 1 g 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).

18 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 S M g g pointwise, we now know that S M g g uniformly since for small enough neighborhoods, the blip is always outside the boundaries. (revisit square wave animation)

19 4. Gibbs Phenomenon: General Principles 1 Kernels And Convolutions Kernels: Def n : (Good Kernel) A family {K n } n N real-valued integrable functions on the circle is a family of good kernels if the following hold: (i) n N, 1 2π K n(θ) dθ = 1. (ii) M > 0 such that n N, (iii) δ > 0, δ θ <π K n(θ) dθ M. K n(θ) dθ 0 as n.

20 4. Gibbs Phenomenon: General Principles 1 Kernels And Convolutions Kernels: Def n : (Good Kernel) A family {K n } n N real-valued integrable functions on the circle is a family of good kernels if the following hold: (i) n N, 1 2π K n(θ) dθ = 1. (ii) M > 0 such that n N, (iii) δ > 0, δ θ <π Dirichlet: D N (θ) = K n(θ) dθ M. K n(θ) dθ 0 as n. n N e inθ

21 4. Gibbs Phenomenon: General Principles 1 Kernels And Convolutions Kernels: Def n : (Good Kernel) A family {K n } n N real-valued integrable functions on the circle is a family of good kernels if the following hold: (i) n N, 1 2π K n(θ) dθ = 1. (ii) M > 0 such that n N, (iii) δ > 0, δ θ <π Dirichlet: D N (θ) = D N (θ) dθ = K n(θ) dθ M. K n(θ) dθ 0 as n. n N n N e inθ e inθ dθ

22 4. Gibbs Phenomenon: General Principles 1 Kernels And Convolutions Kernels: Def n : (Good Kernel) A family {K n } n N real-valued integrable functions on the circle is a family of good kernels if the following hold: (i) n N, 1 2π K n(θ) dθ = 1. (ii) M > 0 such that n N, (iii) δ > 0, δ θ <π Dirichlet: D N (θ) = D N (θ) dθ = K n(θ) dθ M. K n(θ) dθ 0 as n. n N n N e inθ e inθ dθ Fejér: F N (θ) = [D 0 (θ) + D 1 (θ) D N 1 (θ)]/n

23 4. Gibbs Phenomenon: General Principles 1 Kernels And Convolutions Kernels: Def n : (Good Kernel) A family {K n } n N real-valued integrable functions on the circle is a family of good kernels if the following hold: (i) n N, 1 2π K n(θ) dθ = 1. (ii) M > 0 such that n N, (iii) δ > 0, δ θ <π Dirichlet: D N (θ) = D N (θ) dθ = K n(θ) dθ M. K n(θ) dθ 0 as n. n N n N e inθ e inθ dθ Fejér: F N (θ) = [D 0 (θ) + D 1 (θ) D N 1 (θ)]/n Recall: Dirichlet Kernel is NOT a good kernel and Fejér Kernel is.

24 4. Gibbs Phenomenon: General Principles 2 Kernels And Convolutions Cont. Convolution: Def n : The convolution of two functions, f and g is given by: (f g)(θ) = 1 2π f(y)g(θ y) dy (1)

25 4. Gibbs Phenomenon: General Principles 2 Kernels And Convolutions Cont. Convolution: Def n : The convolution of two functions, f and g is given by: S M f = (D M f)(θ) (f g)(θ) = 1 2π f(y)g(θ y) dy (1)

26 4. Gibbs Phenomenon: General Principles 2 Kernels And Convolutions Cont. Convolution: Def n : The convolution of two functions, f and g is given by: S M f = (D M f)(θ) (f g)(θ) = 1 2π f(y)g(θ y) dy (1) Def n : σ M f(θ) := [S 0 f(θ) + S 1 f(θ) S M 1 f(θ)]/m

27 4. Gibbs Phenomenon: General Principles 2 Kernels And Convolutions Cont. Convolution: Def n : The convolution of two functions, f and g is given by: S M f = (D M f)(θ) (f g)(θ) = 1 2π f(y)g(θ y) dy (1) Def n : σ M f(θ) := [S 0 f(θ) + S 1 f(θ) S M 1 f(θ)]/m From a previous homework, we also know that F M f = σ M f

28 4. Gibbs Phenomenon: General Principles 2 Kernels And Convolutions Cont. Convolution: Def n : The convolution of two functions, f and g is given by: S M f = (D M f)(θ) (f g)(θ) = 1 2π f(y)g(θ y) dy (1) Def n : σ M f(θ) := [S 0 f(θ) + S 1 f(θ) S M 1 f(θ)]/m From a previous homework, we also know that F M f = σ M f F M f is of sort an averaging of the Dirichlet kernels, or more importantly, the partial Fourier sums.

29 4. Gibbs Phenomenon: General Principles 3 The Problem is Quite General Brocher showed Gibbs Phenomenon occurs with any 2π-periodic piecewise-continuous function with a jump discontinuity Overshoot is proportional to height of jump: Square Wave: Overshoot = k(2) k = Overshoot/Undershoot given by (f(x + d ) f(x d ) where x d is point of discontinuity Suggests an overshoot of (2π) for a maximum height of Thus, need a way to resolve this for modeling of periodic phenomena to be practical Using Fejér Kernel averages Dirichlet kernels and could therefore be a way of eliminating this phenomenon animation experiment

30 5. Conclusion 1 Gibbs Phenomenon observed in all 2π-periodic functions with a jump discontinuity In some cases, it is quite basic to show it s persistence analytically Causes nastiness in approximations of functions by Fourier series, but is easily remedied with averaging using Fejér Kernel