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

12

f(x)einx dx

einxdenotes 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}nN converges pointwise to f on the set Dif > 0 and x D, N N such that |fn(x) f(x)| < n N .

Defn: Uniform Convergence of Functions:

A series of functions {fn}nN converges uniformly to f on the set Dif > 0, N N such that |fn(x) f(x)| < n N and x D.

4 3 2 1 0 1 2 3 41.5

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.

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

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 Kelvins 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, Michelsons machineclosely approximated the sawtooth function except for two blips nearthe 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 4 alternately above and below it,joined by perpendiculars which are themselves part of the locus the

equation overshoots 4 by a distance of:12

sin(x)x dxAlso 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

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

0

sin(u)

udu.

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

3. Gibbs Phenomenon: An Example 1

Consider the square wave function given by:

g(x) :=

{1 if x < 01 if 0 x <

Goal: Analytically prove that Gibbs Phenomenon occurs at jumpdiscontinuities of this 2-periodic function

3. Gibbs Phenomenon: An Example 2

Using the definition an :=12

f(x)einx dx, basic Calculus yields

SNf(x) =

Nn=N

anein =

1

|n|Nn is odd

(2

n

)ein

i

=4

Mm=1

sin((2m 1))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))2m 1

=

x0

cos((2m 1)) d

which yields the following identity:

S2M1g(x) =4

Mm=1

sin((2m 1)x)2m 1

=4

Mm=1

x0

cos((2m 1)) d

=4

x0

Mm=1

cos((2m 1)) d

3. Gibbs Phenomenon: An Example 4

Now using the identity

Mm=1

cos((2m 1)x) = sin(2Mx)2 sin(x)

we get the following:

S2M1g(x) =4

x0

Mm=1

cos((2m 1)) d

=4

x0

sin(2M)

2 sin()d

3. Gibbs Phenomenon: An Example 5

At this point, using basic analysis of the derivative of S2M1g(x), we seethat there are local maximum and minimum values at

xM,+ =

2Mand xM, =

2MHere, using the substitution u = 2M and the fact that for u 0,sin(u) u, we get that for large M

S2M1g(xm,+) 2

0

sin()

d

and

S2M1g(xm,) 2

0

sin()

d

Further, the value 2

0

sin()

d 1.17898. Have shown that

M >> 1, M N, x = xM,+ such that S2M1g(xm,+) 1.17898. is always a blip near the jump discontinuity, namely at x = 2M ,where the value S2M1g(x) is not near 1!

3. Gibbs Phenomenon: An Example 6

Have shown analytically that a blip persists for all S2M1g near thejump discontinuity.

Demonstrated again by square wave

We can also see the same behavior in the 2-periodic sawtoothfunction given by f(x) = x on the interval [, ).

3. Gibbs Phenomenon: An Example 6

Have shown analytically that a blip persists for all S2M1g near thejump discontinuity.

Demonstrated again by square wave

We can also see the same behavior in the 2-periodic sawtoothfunction given by f(x) = x on the interval [, ).

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)

4. Gibbs Phenomenon: General Principles 1

Kernels And ConvolutionsKernels:

Defn: (Good Kernel) A family {Kn}nN real-valued integrablefunctions on the circle is a family of good kernels if the following hold:

(i) n N, 12

Kn() d = 1.

(ii) M > 0 such that n N,

|Kn()| d M .

(iii) > 0,

||

4. Gibbs Phenomenon: General Principles 1

Kernels And ConvolutionsKernels:

Defn: (Good Kernel) A family {Kn}nN real-valued integrablefunctions on the circle is a family of good kernels if the following hold:

(i) n N, 12

Kn() d = 1.

(ii) M > 0 such that n N,

|Kn()| d M .

(iii) > 0,

||

4. Gibbs Phenomenon: General Principles 1

Kernels And ConvolutionsKernels:

Defn: (Good Kernel) A family {Kn}nN real-valued integrablefunctions on the circle is a family of good kernels if the following hold:

(i) n N, 12

Kn() d = 1.

(ii) M > 0 such that n N,

|Kn()| d M .

(iii) > 0,

||

4. Gibbs Phenomenon: General Principles 1

Kernels And ConvolutionsKernels:

Defn: (Good Kernel) A family {Kn}nN real-valued integrablefunctions on the circle is a family of good kernels if the following hold:

(i) n N, 12

Kn() d = 1.

(ii) M > 0 such that n N,

|Kn()| d M .

(iii) > 0,

||

4. Gibbs Phenomenon: General Principles 1

Kernels And ConvolutionsKernels:

Defn: (Good Kernel) A family {Kn}nN real-valued integrablefunctions on the circle is a family of good kernels if the following hold:

(i) n N, 12

Kn() d = 1.

(ii) M > 0 such that n N,

|Kn()| d M .

(iii) > 0,

||

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)() = 12

f(y)g( y) dy (1)

SMf = (DM f)()

Defn: Mf() := [S0f() + S1f() + ...+ SM1f()]/M

From a previous homework, we also know that FM f = Mf FM f is of sort an averaging of the Dirichlet kernels, or moreimportantly, the partial Fourier sums.

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)() = 12

f(y)g( y) dy (1)

SMf = (DM f)()

Defn: Mf() := [S0f() + S1f() + ...+ SM1f()]/M