Degree Project

The Gibbs Phenomenon and its Resolution

Author: Carolyn Oddy Supervisor: Börje Nilsson Examiner: Joachim Toft Date: 2015-06-17 Course Code: 2MA11E Department Of Mathematics

AbstractIt is well known that given an arbitrary continuous and periodic function f(x), it

is possible to represent it as a Fourier series. However, attempting to approximate adiscontinuous or non periodic function, using a Fourier series, yields very poor results.Large oscillations and overshoots appear around the points of discontinuity. Regard-less of the number of terms that are included in the series, these overshoots do notdisappear, they simply move closer to the point of discontinuity. This is known as theGibbs phenomenon.

In the 1990’s David Gottlieb and Chi-Wang Shu introduced a new method, en-titled the Gegenbauer procedure, which completely removes the Gibbs phenomenon.We will review their method, as well as present a number of examples to illustrateits effect. Going one step further we will discover that not all orthogonal polynomialsmay be treated equal in terms of this Gegenbauer procedure. When replacing Gegen-bauer polynomials with Chebyshev or Legendre polynomials, it appears as though theinability to vary λ ultimately makes them ineffective.

1 Introduction

The ability to represent a discontinuous function as an infinite sum of sines andcosines, has long been known in the field of Mathematics [4]. In fact it was alreadyknown to Euler, who in the middle of the 18th century presented the following sum:

∞∑k=1

sin(kx)

k=

1

2(π − x).

Growing on this idea, in 1807, for the purpose of solving heat equations, JeanBaptiste Fourier [10] proposed that it was possible to represent an arbitrary functionf(x) using a trigonometric series of the form

a02

+

∞∑k=1

(ak sin(kx) + bk cos(kx)).

This is what we recognize as a Fourier series today.In Section 2, where greater detail surrounding Fourier series as well as a number

of examples are given, it is explained that Fourier series can be used very effectivelyto approximate smooth, periodic functions. However in the event that a Fourier seriesis used to approximate a discontinuous function, it does very poorly. Even away fromthe point of discontinuity, the convergence of the Fourier series is slow and a largeovershoot around this point can be found.

By 1848, this nonuniform convergence of a Fourier series for discontinuous functionwas being analyzed by Henry Wilbraham. This would be the start of identifying whatwe know as the Gibbs phenomenon [9]. However his work went largely unnoticedand was quickly forgotten. It wasn’t until 50 years later, when a debate sparkedover a publication by Albert Michelson, in the British journal Nature, was the Gibbsphenomenon truly recognized [4].

This phenomenon as the name suggests, was identified by Willard Gibbs. In hispublication, Michelson pointed out that constructing the function f(x) = x using aFourier series proved difficult. Gibbs correctly showed that the large oscillations, foundspecifically around the points of discontinuity, do not decay but rather tend to a fixednumber [4]. This will be discussed further in Section 3.

The Gibbs phenomenon, at a quick glance, appears to imply that it is almost im-possible to obtain accurate local information about discontinuous or piecewise smoothfunctions using a Fourier series. This poses a far reaching problem, as there are manyphysical phenomena that are represented by piecewise smooth functions. For example,

1

there are many problems in aerospace engineering that involve fluid flows containingshockwaves. These shockwaves can be interpreted as discontinuities in the pressurefield. In MRI imaging the Gibbs phenomenon can also cause artefacts to appear in thescan that aren’t in fact there. Numerical weather predictions and data compressioncan also be affected.

There are a number of methods in existence that help to mitigate the effects ofthe Gibbs phenomenon, Filters [4] being the most well known. However they are onlysuccessful in enhancing the accuracy away from the point of discontinuity, and areunable to entirely remove the effect of the Gibbs phenomenon.

In Section 4, we will introduce a method created by David Gottlieb and Chi-WangShu in the 1990’s. This method, termed the Gegenbauer procedure or Gegenbauerseries Approximation, allows for the total removal of the Gibbs phenomenon. It isgiven in detail in [4]. By re-expanding the truncated Fourier series with Gegenbauerpolynomials they are able to obtain exponential accuracy in the maximum norm inany interval of analyticity. In Section 4 we will also give a few examples that illustratethe removal of this phenomenon.

Gottlieb and Shu, together with a number of their colleagues, have published aseries of papers, [4], [5], [6], [7], [8], surrounding the Gegenbauer procedure. Onething they seem to neglect to touch on, is exactly what characteristics of Gegenbauerpolynomials make them appropriate for this method. In finishing, we will examineif it is possible to replace Gegenbauer polynomials with other orthogonal polynomi-als, specifically Chebyshev and Legendre polynomials, and still obtain exponentialaccuracy.

The purpose of this article is to provide a basic understanding of what a Fourierseries and the Gibbs phenomenon are, and also the complications that the phenomenonmay cause. We will review the method created by Gottlieb and Shu as well as analyzea number of examples in order to illustrate its effect. It is hoped finally that byreplacing Gegenbauer polynomials with other orthogonal polynomials, that a greaterunderstanding of the specifics that allow their method to work, will be obtained.

Before we can discuss all of this, let us start from the beginning and introduce indetail, exactly what a Fourier series is.

2 Introducing Fourier Series

An arbitrary function f(x) defined everywhere in [−c, c], may be represented as

f(x) ≈ a02

+

∞∑n=1

(an cos

(nπxc

)+ bk sin

(nπxc

)), (2.1)

where the coefficients an and bn are given by

an =1

c

∫ c

−cf(x) cos

(nπxc

)dx,

and

bn =1

c

∫ c

−cf(x) sin

(nπxc

)dx.

The series given by (2.1) is termed a Fourier series [2]. The coefficients an and bnare therefore known as the Fourier coefficients.

Since cos(nπxc

) + i sin(nπxc

) = einπxc , another way to represent a Fourier series is

as follows [1],

f(x) ≈∞∑

n=−∞

fneinπxc ,

2

where the Fourier coefficient fn is

fn =1

c

∫ c

−cf(x)e−i

nπxc dx.

By referring to a truncated or partial Fourier series, denoted fN (x), it is meantthat the series is only taken to some value N instead of to infinity, i.e we truncate theseries.

Provided that the arbitrary function f(x) is smooth and periodic, then a Fourierseries is a very good way to accurately reconstruct the function. In fact, it is knowthat if f(x) is analytic and periodic, then the Fourier series converges to the functionexponentially fast [4]. That is that,

max−c≤x≤c

|f(x)− fN (x)| ≤ e−αN , α > 0.

However, if f(x) is non periodic, or discontinuous, then fN (x) yields a very poorapproximation of the function. This is possible to observe in the following two exam-ples.

2.1 Example One

As a first example, let us calculate the truncated Fourier series for the function

f(x) = x, −π ≤ x ≤ π.

Since f(x) = x is an odd function, i.e f(x) = −f(−x), it is not difficult to make thequick observation that the Fourier coefficient an will be equal to zero. What remainsis to calculate bn.

bn =1

π

∫ π

−πf(x) sin

(nπxπ

)dx

=2

π

∫ π

0

x sin(nx)dx.

Using integration by parts we obtain the following:

=2

π

(−[

cos(nx)

nx

]π0

+

∫ π

0

cos(nx)

xdx

).

In taking the integral from 0 to π of the function cos(nx)n

, we obtain a zero, thereforewe are left with,

bn =−2

πn[cos(πn) · π]

=−2

π[cos(πn)] .

For odd values of n, cos(πn) = −1, whereas for even values of n, cos(πn) = 1.Thus, taking into account the negative sign in front of 2 we obtain,

bn =2(−1)n+1

n.

The truncated Fourier series for f(x) = x, where −π ≤ x ≤ π is then given by:

fN (x) = 2

N∑n=1

(−1)n+1 sin(nx)

n.

In the figures below, it is possible to see the truncated Fourier series for variousvalues of N . Take special note of the large overshoots that occur around the endpoints, and how they appear to never decrease in height. This will be discussed in thefollowing section.

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Figure 2.1: fN (x), where N = 5 Figure 2.2: fN (x), where N = 10

Figure 2.3: fN (x), where N = 20

2.2 Example Two

As a second example, let us calculate the Fourier series for the square wave function

f(x) =

{1 0 ≤ x ≤ π−1 −π ≤ x < 0

.

This is again, an odd function. We are therefore only required to calculate bn.

bn =1

π

∫ π

π

f(x) sin(nπx

π)dx

=2

π

∫ π

0

sin(nx)dx

=−2

πn[cos(nx)]π0

=−2

πn[cos(πn)− 1]

If n is an even number, then the value within the brackets goes to zero. Howeverif n is odd, then it is equal to 2. Hence, we only need to consider the Fourier series forodd values of n. For simplicity, we may rewrite the coefficient bn and the truncatedFourier series as follows.

b2n−1 =4

π(2n− 1)

fN (x) =4

π

N∑n=1

sin((2n− 1)x)

2n− 1

Again, the figures of the truncated Fourier series for different values of N maybe found below. In the same way as the previous example, one may notice largeoscillations or overshoots around the points of discontinuity.

4

Figure 2.4: fN (x), where N = 5 Figure 2.5: fN (x), where N = 10

Figure 2.6: fN (x), where N = 20

3 Introduction of The Gibbs Phenomenon

To understand the concept of the Gibbs phenomenon, let us return to two of theexamples we dealt with in the previous section: the square wave, and the line f(x) = x,in the interval −π ≤ x ≤ π.

Figure 3.1: Fourier series of f(x) = x,where N = 20

Figure 3.2: Fourier series of squarewave, where N = 20

In both examples it is possible to see large deviations near the points of disconti-nuity of the Fourier series. This is known as the Gibbs Phenomenon.

Theorem 3.1 Let f be a piecewise smooth function with a period of 2π. Suppose thatx0 is a point of discontinuity, then the deviation of the Fourier series caused by theGibbs Phenomenon will be approximately equal to 0.9(f(x+0 )− f(x−0 )). That is, it willbe approximately equal to 9% of the length of the jump discontinuity. [9]

We continue by taking a closer look at the piecewise smooth function

f(x) =

{−1 −π < x < 0

1 0 < x < π,

5

which, as we have already seen has the partial Fourier series

fN (x) =4

π

N∑n=1

sin((2n− 1)x))

2n− 1. (3.1)

=4

π

(sin(x) +

sin(3x)

3+ ...+

sin((2N − 1)x)

2N − 1

). (3.2)

In order to gain a better understanding of the behaviour of the Fourier seriesaround the point of discontinuity, let us start by differentiating (3.2). This allows usto find the location of the Fourier series’ critical points. For more in-depth calculations,please refer to Appendix 7.1.

f ′N (x) =4

π(cos(x) + cos(3x) + ...+ cos((2N − 1)x))

=4

π

N∑n=1

cos((2n− 1)x)

=2 sin(2Nx)

π sin(x).

By setting f ′N to zero, we find that the critical points of the function are givenwhen

2Nx = ±π,±2π,±3π, ...

We are however, only interested in two of these points. The local maximum founddirectly to the right of the origin, and the local minimum found to the left. We willfocus on the behaviour of the point to the right hand side. That is when

x =π

2N.

Now that we know where the maximum value of the overshoot caused by the Gibbsphenomenon is, let us take a closer look at the value of the Fourier series at that point.

fN( π

2N

)=

4

π

N∑n=1

sin((2n− 1)( π

2N))

2n− 1

To understand the behaviour of the Fourier series for large values of N, we mayconsider the integral

g(x) =

∫ π

0

sin(x)

x,

which can be approximated using a Riemann sum. By partitioning the interval [0, π]into N equal equal parts each of which have a length of π

N, we are able to gain a good

estimate of the curve sin(x)x

for large values of N . We have

∫ π

0

sin(x)

x= limN→∞

sin(π2N

)π2N

· πN

+sin(3πN

)3πN

· πN

+ ...+sin(

(2n−1)π2N

)(2n−1)π

2N

· πN

= limN→∞

N∑n=1

sin((2n− 1)( π

2N))

2n− 1

=π

2limN→∞

fN( π

2N

)limN→∞

fN (π

2N) =

2

π

∫ π

0

sin(x)

x

≈ 1.17898

6

The length of the jump discontinuity in this example is equal to two. Therefore,the overshoot of the Fourier series caused by the Gibbs phenomenon is approximatelyequal to 0.17898/2 ≈ 0.9 as stated in Theorem 3.1.

Regardless of the size of N , i.e the number of terms that are taken in the Fourierseries, this overshoot never decays. It simply moves closer to the point of disconti-nuity. There are a large number of methods in existence that attempt to resolve theGibbs phenomenon, such as filters. However they are only successful in enhancing theaccuracy of the Fourier series away from the discontinuity.

In the following sections, we will examine a method by David Gottlieb and Chi-Wang Shu that is outlined in [4]. Their procedure, which employs the use of Gegen-bauer polynomials, allows us to obtain exponential accuracy in the maximum norm inany interval of analyticity. That is, to completely remove the Gibbs phenomenon.

4 Resolving the Gibbs Phenomenon Using theGegenbauer Procedure

Using the Gegenbauer procedure, which will be outlined in the following sections, itis possible to completely resolve the Gibbs Phenomenon. This method was developedby David Gottlieb and Chi-Wang Shu. It is outlined in detail in [4]. In short, usingthe knowledge of the Fourier expansion coefficients of a discontinuous but piecewiseanalytic function, it is possible to construct a new rapidly converging series based onanother expansion coefficient.

As the name suggests, this method makes use of Gegenbauer polynomials, denotedCλk (x). Gegenbauer polynomials are orthogonal over the range x ∈ [−1, 1], with theweight function (1 − x2)λ−1/2. For a full description of their properties, refer toAppendix 7.2

Depending on the characteristics of the problem at hand, resolving the Gibbs phe-nomenon requires us to employ one of two slightly different methodologies. We willexamine the Gegenbauer procedure for the two following problems.

Problem 1: Given 2N + 1 Fourier coefficients, which we denote fn(x), for −N ≤n ≤ N , of an analytic but non periodic function f(x) in −1 ≤ x ≤ 1, construct ac-curately the point values of a function. An analytic function is defined as a functionthat can be locally given by a convergent powerseries.

Problem 2: Given 2N + 1 Fourier coefficients fn(x), for −N ≤ n ≤ N , of apiecewise analytic function f(x) in −1 ≤ x ≤ 1 construct accurately the values of thefunction in any subinterval [a, b] ⊂ [−1, 1] that does not include a discontinuity.

4.1 Gegenbauer Procedure on the Interval [-1,1]

As it is implied in Problem 1, to construct the Gegenbauer approximation of agiven function f(x), we must know its respective Fourier expansion coefficient, fn(x).With the Fourier expansion coefficient, it is not difficult to then calculate the truncatedFourier series approximation fN (x).

When attempting to approximate a function which is analytic over the whole in-terval [-1,1], using the Gegenbauer procedure, there are two main steps.

Step one: Compute the first m+ 1 Gegenbauer coefficients:

gλk =1

hλk

∫ 1

−1

(1− x2

)λ− 12 fN (x)Cλk (x)dx, 0 ≤ k ≤ m, (4.1)

7

where hλk are the normalization constants, and Cλk (x) are the Gegenbauer polynomials.A full description of both can be found in Appendix 7.2.

Step two: Calculate the Gegenbauer series approximation of f(x).

f(x) ≈m∑k=0

gλkCλk (x).

4.1.1 Numerical Example

Let us now approximate the function f(x) = x using the Gegenbauer procedureoutlined above. As this function was used originally in 1898 to demonstrate theGibbs phenomenon, it seems natural that we should attempt to resolve the Gibbsphenomenon using f(x) = x as an example.

As any polynomial can be written as a convergent power series, we know that thefunction

f(x) = x, −1 ≤ x ≤ 1

is analytic over the whole interval [-1,1].The Fourier series, fN (x) of f(x) is given by

fN (x) =−2

π

N∑n=1

(−1)n

nsin(nπx)

Therefore the first m+ 1 Gegenbauer coefficients are given by

gλk =1

hλk

∫ 1

−1

(1− x2

)λ− 12

2

π

N∑n=1

(−1)2n−1

nsin(nπx)Cλk (x)dx, 0 ≤ k ≤ m.

The Gegenbauer approximation of the function f(x) = x on the interval [−1, 1] isthen the sum

m∑k=0

gλkCλk (x).

In looking at Figure 4.1 below, where λ = m = 1 and N = 4, we can see thatthe Gegenbauer series approximation, with only two terms, already yields a far betterapproximation than strictly the Fourier series with four terms.

The Gegenbauer approximation is most effective if λ and m grow linearly with N .This will be discussed further in a later section. If we then let λ = m = 2 and N = 8,it can be seen in Figure 4.2 that a very accurate approximation of f(x) is obtained bythe Gegenbauer series.

8

Figure 4.1: Gegenbauer series approximation of f(x) = x. Here N = 4, λ =1,m = 1.

Figure 4.2: Gegenbauer series approximation of f(x) = x. Here N = 8, λ = 2,m = 2.

9

4.2 Gegenbauer Procedure on the Interval [a,b]⊂[-1,1]

Attempting to resolve the Gibbs phenomenon, using the procedure outlined inthe previous section, is not possible if f(x) is piecewise analytic. However, using themore general method that follows, it is possible to recover the function values fromthe Fourier series, if f(x) is analytic on sub-intervals [a, b] ⊂ [−1, 1].

Step one: Given a function f(x) which is analytic on the interval [a, b], define alocal variable ξ such that f(x) is now analytic on the interval −1 ≤ ξ ≤ 1:

ξ = ξ(x) =x− δε

, i.e x = x(ξ) = εξ + δ,

where

ε =b− a

2, and δ =

b+ a

2.

Step two: Calculate the first m + 1 Gegenbauer coefficients gλε (l) of the Fourierseries fN (εξ + δ). Using this notation, the variable l in gλε (l) holds the place of theindex k in (4.1)

gλε (l) =1

hλl

∫ 1

−1

(1− ξ2

)λ− 12 Cλl (ξ) fN (εξ + δ)dξ. (4.2)

Step three: Calculate the Gegenbauer series approximation of f(εξ + δ) where−1 ≤ ξ ≤ 1:

f(εξ + δ) ≈m∑l=0

gλε (l)Cλl (ξ) .

Step four: Repeat steps one through three for each interval [a, b] ⊂ [−1, 1] on whichthe function is analytic. To see the Gegenbauer approximation of each respectiveinterval on the same coordinate system set ξ = x−δ

ε.

4.2.1 Numerical Example

Let us use the procedure outlined above, to approximate the function

f(x) =

{−1 −1 ≤ x < 0

1 0 ≤ x ≤ 1.(4.3)

The partial Fourier series of f(x) is given by

fN (x) =4

π

N∑n=1

sin (nπx)

n. (4.4)

As every second term in the series will be zero, we may re-write it as

fN (x) =4

π

N∑n=1

sin ((2n− 1)πx)

2n− 1. (4.5)

The function in (4.3), has a point of discontinuity at x = 0. Therefore we mustredefine the two parts of the function such that they are continuous over the whole

10

interval −1 ≤ ξ ≤ 1. Let us begin by working with the function defined where[a, b] = [0, 1]. We therefore have that

ξ =x− δε

= 2x− 1, i.e that x = εξ + δ =ξ + 1

2,

where

ε =b− a

2=

1

2, and δ =

b+ a

2=

1

2.

In continuing with the outlined method, we find that the firsts m+ 1 Gegenbauercoefficients of the Fourier series fN ( ξ+1

2) are given by

gλε (l) =1

hλl

∫ 1

−1

(1− ξ2

)λ− 12 Cλl (ξ)

4

π

N∑n=1

sin(

(2n− 1)π(ξ+1)2

)2n− 1

dξ, 0 ≤ l ≤ m,

and that the Gegenbauer series approximation of the function

f

(ξ + 1

2

)= 1, −1 ≤ ξ ≤ 1,

is given by

f

(ξ + 1

2

)≈

m∑l=0

gλε (l)Cλl .

In Figure 4.3, it can be seen that the Gegenbauer series approximation is far betterthan that of the Fourier series. In this case the first 10 terms of the Fourier series weretaken ( i.e N = 5 in (4.5) or N = 10 in (4.4)), and m = λ = 3.

Figure 4.3: Gegenbauer series approximation of f( ξ+12 ) = 1. Here N = 5, λ = 3,

m = 3.

11

Now, let us consider the second interval where [a, b] = [−1, 0]. In this case we havethat

ε =b− a

2=

1

2and that, δ =

b+ a

2=−1

2.

Therefore

ξ = ξ(x) =x− δε

= 2x+ 1 and x = x(ξ) = εξ + δ =ξ − 1

2.

As before, it is then necessary to compute the first m+ 1 Gegenbauer coefficientsof fN ( ξ−1

2), and the Gegenbauer series approximation of f( ξ−1

2) = −1. That is, we

calculate

gλε (l) =1

hλl

∫ 1

−1

(1− ξ2

)λ− 12 Cλl (ξ)

4

π

N∑n=1

sin(

(2n− 1) π(ξ−1)2

)2n− 1

dξ,

and

f

(ξ − 1

2

)≈

m∑l=0

gλε (l)Cλk .

In Figure (4.4) we can again see that the Gegenbauer series approximation yields avast improvement over the Fourier series.

Figure 4.4: Gegenbauer series approximation of f( ξ−12 ) = −1. Here N = 5,

λ = 3, m = 3.

In order to view the complete Gegenbauer series approximation of (4.3) on thesame coordinate system, which can be found in Figure 4.5, we do a change of variablesand let ξ = x−δ

εin each respective part. That is, for the Gegenbauer procedure on

the interval [a, b] = [0, 1] we let ξ = 2x + 1, and on the interval [a, b] = [−1, 0] we letξ = 2x+ 1.

12

Figure 4.5: Gegenbauer series approximation of f(x). Here N = 5, λ = 3,m = 3.

5 Truncation and Regularization Errors

The proof that the outlined Gegenbauer procedure provides exponential conver-gence involves showing that the errors caused by expanding the Nth Fourier coefficientinto Gegenbauer polynomials can be made exponentially small.

Let fmN be the Gegenbauer series approximation of the function f(x), given by

fmN =

m∑k=0

gλkCλk (x).

Again, the Gegnbauer polynomials are given by Cλk (x), and the Gegenbauer seriesexpansion coefficients gλk are given by

gλk =1

hλk

∫ 1

−1

(1− x2)λ−12 fN (x)Cλk dx. (5.1)

Also, let fm be the expansion of f(x) into mth-degree Gegenbauer polynomials.That is let

fm(x) =

m∑k=0

fλkCλk (x),

where

fλk =1

hλk

∫ 1

−1

(1− x2

)λ− 12 f(x)Cλk (x)dx. (5.2)

Showing that the error

||f − fmN || = ||f − fm + fm − fmN || ≤ ||f − fm||+ ||fm − fmN ||

13

can be made exponentially small, is possible by showing that each term on the righthand side can be made exponentially small respectively.

Here the first term on the right hand side is the regularization error, and the secondterm is the truncation error. Also by the norm || · ||, we mean the maximum normover [−1, 1]. It is also important in this section, that we make the assumption thatthe function f(x) is L1. For a definition of Lp functions, please refer to Appendix 7.3.All theorems, definitions and assumptions are taken and based off of those presentedin [4].

5.1 Truncation Error

The truncation error is incurred as we calculate the first m+ 1 Gegenbauer coef-ficients. It is the measure of the distance between the true Gegenbauer expansion off(x) and its approximation based on the truncated Fourier series. The definition ofthe truncation error is given below.

Definition 5.1 Let f(x) be an L1 function that is analytic on the interval [−1, 1].The truncation error may then be defined as:

||fm − fmN || = max−1≤x≤1

∣∣∣∣∣m∑k=0

(fλk − gλk

)Cλk (x)

∣∣∣∣∣ .Here fλk are the Gegenbauer coefficients of the original function f(x) and gk are theGegenbauer coefficients of the Fourier series. They are defined in (5.2) and (4.1)respectively.

If however, f(x) is not analytic on the whole interval [−1, 1], but in a subinterval[a, b] ⊂ [−1, 1], we may introduce the same change of variable as in section 4.2. Inthat way, the function f(x) is analytic on the interval −1 ≤ ξ ≤ 1. We let

x = εξ + δ, i.e ξ =x− δε

,

where

ε =b− a

2, and δ =

a+ b

2.

The function f(x), therefore has a Gegenabauer expansion in the subinterval [a, b]given by

f(εξ + δ) =

m∑l=0

fλε (l)Cλl (ξ).

The Gegenbauer coefficients fλε (l) are defined by

fλε (l) =1

hλl

∫ 1

−1

(1− ξ2)λ−12Cλl (ξ)f(εξ + δ). (5.3)

Let us now restate the definitions for the truncation error of a function that is analyticon an interval [a, b] ⊂ [−1, 1]. As in the previous section, the variable l in fλε (l) andgλε (l) holds the place of the index k in fλk and gλk .

14

Definition 5.2 Let f(x) be an L1 function, that is analytic on the interval [a, b] ⊂[−1, 1]. The truncation error may then be defined as:

TE(λ,m,N, ε) = max−1≤ξ≤1

∣∣∣∣∣m∑l=0

(fλε (l)− gλε (l)

)Cλl (ξ)

∣∣∣∣∣ , (5.4)

where fλε (l) are given by (5.3) and gλε (l) are given by (4.2)

We may now summarize the conditions on the variables that allow us to achievean exponentially small truncation error.

Theorem 5.1 Let the truncation error be defined by (5.4). Let λ = αεN and m =βεN with α and β as constants. Then,

TE(αεN, βεN,N, ε) ≤ AqεNT ,

where A grows at most as a fixed-degree polynomial of N , and qT is given by

qT =(β + 2α)β+2α

2αααββ.

If α and β are chosen such that α = β ≤ 227

then 0 < qT < 1

Details of the proof can be found in [4] and [5].

5.2 Regularization Error

The error that is incurred while using a finite (first m + 1 terms) Gegenbauerexpansion to approximate an L1 function f(x), which is analytic in [-1,1], is labeledthe regularization error. It yields the following definition.

Definition 5.3 The regularization error is defined by

RE(λ,m) = ||f − fm|| = max−1≤x≤1

∣∣∣∣∣f(x)−m∑k=0

fλkCλk

∣∣∣∣∣ . (5.5)

Here fλk is again defined by (5.2)

Before we may summarize the conditions on the variables that yield an exponen-tially small regularzation error, we must present some standard assumptions that aresatisfied by analytic functions in [−1, 1].

Assumption 5.1 There exists a constant 0 ≤ r0 < 1 and an analytic extension off(x) onto the elliptic domain

Dρ ={z : z =

1

2

(reiθ +

1

re−iθ

)0 ≤ θ ≤ 2π, r0 ≤ r ≤ 1

}.

Assumption 5.2 Given an analytic function f(x) on [−1, 1] there exists constantsρ ≥ 1 and C(ρ) such that for every k ≥ 0,

max−1≤x≤1

∣∣∣∣dkfdxk(x)

∣∣∣∣ ≤ C(ρ)k!

ρk.

Here, ρ is the distance from [−1, 1] to the nearest singularity of f(x) in the complexplane.

15

Depending on which assumption is used, it is possible to obtain two slightly differ-ent estimates for the regularization error. Both will be presented and briefly discussed.

Using Assumption 5.1, we may obtain the following estimate.

Theorem 5.2 Given an L1 function f(x) which is analytic on [−1, 1], the regulariza-tion error, defined by (5.5), satisfies:

RE(γm,m) ≤ Amqm, (5.6)

where q is given by

q =(1 + 2γ)

1+2γ2

(2γ)γr.

If λ = γm, where γ > 0 and γ is small enough, then 0 < q < 1.

Details of the proof can be found in [4] and [5]. Using Assumption 5.2 instead,yields a slightly different estimate presented below.

Theorem 5.3 Given an L1 function f(x) which is analytic on [−1, 1], the regulariza-tion error, defined by (5.5), satisfies:

RE(γm,m) ≤ Aqm. (5.7)

Here q is given by

q =(1 + 2γ)1+2γ

ρ21+2γγγ(1 + γ)1+γ.

Assuming that λ = γm, where γ > 0, then since ρ ≥ 1, 0 < q < 1.

Details of the proof may again be found in [4] and [5].Following the short discussing in [4] it is the opinion of the writer that the estimate

of q given by (5.7) is slightly more useful. Although making the assumption that ρ ≥ 1restricts the class of analytic functions to which the estimate can apply, and in certaincircumstances the estimate given by (5.6) can be more accurate than that given by(5.7), (5.7) will always provide a useable estimate. In the event that λ is chosen tobe too large, (say γ = 2.5, which is given as an example by Gottlieb and Shu), then(5.6) produces a useless estimate where q > 1, while (5.7) still gives an estimate whereq < 1.

In the event that the function f(x) is only analytic on subintervals [a, b] ⊂ [−1, 1],we may again introduce the change of variables, such that f(x)=f(εξ + δ) is analyticon the interval −1 ≤ ξ ≤ 1. This may then be translated back to the function f(x)with the suitable factor involving ε.

Let us now restate the definition and Theorem 5.3 involving the regularizationerror for the more general case when f(x) in analytic in [a, b] ⊂ [−1, 1].

Definition 5.4 Given and L1 function f(x) which is analytic on an interval [a, b] ⊂[−1, 1] the regularization error may be defined as

RE(γm,m, ε) = max−1≤ξ≤1

∣∣∣∣∣f(εξ + δ)−m∑l=0

fλε (l)Cλl (ξ)

∣∣∣∣∣ , (5.8)

where fλε (l) is given by (5.2).

16

Theorem 5.4 Given an L1 function f(x) which is analytic on [a, b] ⊂ [−1, 1], theregularization error defined by (5.8) satisfies:

RE(γm,m) ≤ AqmR .

Here q is given by

qR =ε(1 + 2γ)1+2γ

ρ21+2γγγ(1 + γ)1+γ.

Assuming that λ = γm, where γ > 0, then since ρ ≥ 1, 0 < q < 1.

Details regarding the proof can again be found in [4] and [5].

5.3 The Resolution of the Gibbs Phenomenon

We may now combine the results from the previous two subsections. In doing so,we obtain the following.

Theorem 5.5 Consider and L1 function f(x) on [−1, 1], which is analytic in a subin-terval [a, b] ⊂ [−1, 1], and satisfies Assumption 5.1 and Assumption 5.2. Assume thateither the first 2N + 1 Fourier coefficients or the first m + 1 Gegenbauer coefficientsbased on the Gegenbauer polynomials Cλk (x), are known. Let gλε (l), 0 ≤ l ≤ m, be theGegenbauer expansion coefficients, based on the subinterval [a, b], of the Fourier partialsum fN (x) defined by (4.1). Then for λ = m = βεN where β ≤ 2

27, we have

||f − fmN || = max−1≤ξ≤1

∣∣∣∣∣f(εξ + δ)−m∑l=1

gλε (l)Cλl (ξ)

∣∣∣∣∣ ≤ A(qεNT + qεNR

),

where 0 ≤ qT ≤ 1 and 0 ≤ qR ≤ 1.

As was stated previously, this theorem relies on the fact that the truncation andregularization errors can individually be made exponentially small. Since qT and qRare both less than 1, as we increase the size of N the error does decrease exponentially.

In conclusion the Gegenbauer procedure, created by Gottlieb and Shu shows thatit is possible to completely remove the effect of the Gibbs phenomenon and obtainuniform exponential accuracy of a piecewise analytic function given its first 2N + 1Fourier expansion coefficients.

6 The Possibility of Using Other Orthogonal Poly-nomials

One topic that Gottlieb and Shu neglect to touch on in a series of their publicationsregarding this Gegenbauer procedure, is exactly what characteristics of Gegenbauerpolynomials make them appropriate to remove the Gibbs phenomenon. This leavesopen the question: do other orthogonal polynomials on [−1, 1] hold the same propertiesthat would allow them to be used instead of Gegenbauer polynomials?

In short, the answer is ’no’. However to understand why, let us consider a few exam-ples and discuss them briefly. Both Legendre polynomials and Chebyshev polynomialsare orthogonal on [−1, 1], making them appropriate candidates. A full definition oforthogonal, Legendre and Chebyshev polynomials can be found in Appendix7.2.

17

6.1 Using Chebyshev Polynomials Instead of GegenbauerPolynomials

In order to see the results of attempting to resolve the Gibbs phenomenon usingChebyshev polynomials, let us again use the function f(x) = x as an example.

When replacing the Gegenbauer polynomials by Chebyshev polynomials in the pro-cedure outlined in Section 4, keep in mind that we also replace the respective weightfunction and normalization constant. We then carry through the familiar procedurebelow.

Step one: Compute the first m+ 1 Chebyshev coefficients of fN (x):

gk =1

hk

∫ 1

−1

(1− x2)−12 fN (x)Tk(x), 0 ≤ k ≤ m,

where the Chebyshev polynomials and normalization constants are given by Tk(x) andhk respectively. As we have previously seen, the partial Fourier series fN (x) is givenby

fN (x) =−2

π

N∑n=1

(−1)n

nsin(nπx).

Step two: Construct the Chebyshev series

m∑k=0

gkTk(x).

As can be seen in Figures 6.1, 6.2, 6.3 and 6.4, using Chebyshev polynomialsinstead of Gegenbauer polynomials gives, relatively speaking no vast improvementover the partial Fourier series, even for large values of m. In fact, for small values ofm the Chebyshev series does horribly in comparison, regardless of the size of N .

Figure 6.1: Chebyshev series approximation of f(x) = x, where N = 4 andm = 1

18

Figure 6.2: Chebyshev series approximation of f(x) = x, where N = 8 andm = 1

Figure 6.3: Chebyshev series approximation of f(x) = x, where N = 4 andm = 3

19

Figure 6.4: Chebyshev series approximation of f(x) = x, where N = 4 andm = 8

As was stated in Appendix 7.2, Chebyshev polynomials are a special case of Gegen-bauer polynomials where λ is fixed at 0. The theorems in Section 5 that show expo-nential decay of the truncation and regularization error, both require that λ is positive.This requirement provides a convincing explanation as to why Chebyshev polynomialsare a poor choice in attempting to resolve the Gibbs phenomenon.

6.2 Using Legendre Polynomials Instead of GegenbauerPolynomials

The results of repeating the same process for f(x) = x, using Legendre polynomi-als and their respective weight function and normalization constant can be found inFigures 6.5, 6.6, 6.7 and 6.8.

It is possible to see that for m = 1 the Legendre series approximation shows alarge improvement over the truncated Fourier series. As N increases, i.e the numberof Fourier series terms increases, so does the accuracy of the Legendre Approximation.It appears as if it does improve, if not remove the Gibbs phenomenon. If however weincrease m to larger values such as 3 and 8 it can be seen that the estimates, especiallyat the end points, become increasingly worse.

20

Figure 6.5: Legendre series approximation of f(x) = x, where N = 4 and m = 1

Figure 6.6: Legendre series approximation of f(x) = x, where N = 15 andm = 1

21

Figure 6.7: Legendre series approximation of f(x) = x, where N = 4 and m = 3

Figure 6.8: Legendre series approximation of f(x) = x, where N = 4 and m = 8

As is stated in the theorems regarding the removal of the truncation and regulariza-tion errors, it is important that λ = m = βε/N , where β ≤ 2

27. Legendre polynomials

are however a special case of Gegenbauer polynomials where λ = 12. This would again

give us a convincing argument as to why good results are obtained when m = 1 andN is increased, while poor results are obtained for larger values of m.

Following these two examples, it seems as though the ability to vary λ accordingto the choices of m and N contributes to the ability of the Gegenbauer polynomialsto remove the Gibbs phenomenon. Without this ability, specifically in the case of the

22

Legendre polynomials, we are only left with the possibility of scaling N to improvethe accuracy.

Although the Gibbs phenomenon presents many problems across a variety of dis-ciplines, its effects are not insurmountable. In making use of Gegenbauer polynomialsand their related characteristics, Gottlieb and Shu have been successful in develop-ing a method that allows us to completely remove the Gibbs phenomenon. That is,to obtain exponential accuracy when approximating a discontinuous function using aFourier series.

7 Appendix

7.1 Calculation of:

dfN (x)

dx=

N∑n=1

cos ((2n− 1)x) =sin(2Nx)

2 sin(x).

In order to facilitate the full calculation, let us first use the well known trigono-metric identity

2 sinA cosB = sin(A+B) + sin(A−B)

to find the value of

2 sin(x) cos ((2n− 1)x) .

By letting A = x and B = (2n− 1)x, we obtain

2 sin(x) cos ((2n− 1)x) = sin (x+ (2n− 1)x) + sin (x− (2n− 1)x)

= sin(2Nx) + sin (2(1− n)x) .

In continuing, keeping in mind that sin(−x) = − sin(x) we therefore find that

2 sin(x)

N∑n=1

cos ((2n− 1)x) =

N∑n=1

sin(2Nx) + sin (2(1− n)x)

= sin(2x) + sin(0) + sin(4x)− sin(2x) + sin(6x)− sin(4x) + ...

+ sin (2(N − 1)x) + sin (2(1− (N − 1))) + sin(2Nx) + sin (2(1−N)x)

The only term in this series that does not cancel is

sin(2Nx).

Therefore

N∑n=1

cos ((2n− 1)x) =sin(2Nx)

2 sin(x).

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7.2 Orthogonal and Orthonormal Polynomials

A system of polynomials fn(x) is called orthogonal [1] on the interval a ≤ x ≤ bwith respect to a weight function w(x) if∫ b

a

w(x)fn(x)fm(x)dx = δmnhn, n 6= m;n,m = 0, 1, 2....

Here δmn is the Kronecker delta defined by

δmn =

{0, n 6= m

1, n = m

and hn is the norm of the polynomials given by

hn =

∫ b

a

w(x) [fn(x)]2 dx.

If hn = 1, then the set of polynomials are not only orthogonal but orthonormal [12].There are a number of examples of orthogonal polynomials. We will define Gegen-

bauer, Chebyshev (of the first kind), and Legendre polynomials in more detail.

7.2.1 Gegenbauer Polynomials

Gegenbauer Polynomials [4], given by Cλn(x) where λ ≥ 0, are polynomials ofdegree n that satisfy∫ 1

−1

(1− x2)λ−12Cλn(x)Cλm(x)dx = 0, (n 6= m).

Therefore, they are orthogonal on the interval from −1 ≤ x ≤ 1, and their respectiveweight function is

w(x) = (1− x2)λ−12 .

Gegenbauer polynomials may be defined recursively given that,

Cλ0 (x) = 1

Cλ1 (x) = 2λx

Cλn(x) =1

k

[2x(n+ λ− 1)Cλn−1(x)− (n+ 2λ− 2)Cλn−2(x)

].

Gegenbauer polynomials are standardized by

Cλn(1) =Γ(k + zλ)

n!Γ(2λ),

and therefore are not orthonormal, as their norm is given by

hλn =

∫ b

a

w(x)[Cλn(x)

]2dx = π

12Cλn(1)

Γ(λ+ 12)

Γ(λ)(n+ λ).

The gamma function Γ(x) is defined in a later appendix.

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7.2.2 Chebyshev Polynomials of The First Kind

Chebyshev polynomials [1] of the first kind, denoted Tn(x), are a particular caseof Gegenbauer polynomials where λ = 0. Therefore they are also orthogonal on theinterval −1 ≤ x ≤ 1, however their weight function is then

w(x) = (1− x2)−12 .

Chebyshev polynomials, are polynomials of degree n that satisfy∫ 1

−1

(1− x2)λ−12 Tn(x)Tm(x)dx = 0, (n 6= m),

and may be defined recursively by

T0(x) = 1

T1(x) = x

Tn(x) = 2xTn−1 − Tn−2.

They may be standardized by Tn(1) = 1, and again are not orthonormal as their normhn is given by

hn =

{π2, n 6= 0

π n = 0.

7.2.3 Legendre Polynomials

Legendre polynomials [1], denoted Pn(x), are again, a special case of Gegenbauerpolynomials, where λ = 1

2. They are also orthogonal on −1 ≤ x ≤ 1, however their

weight function is given by w(x) = 1.Legendre polynomials are therefore polynomials of degree n that satisfy∫ 1

−1

Pn(x)Pm(x)dx = 0, (n 6= m),

and may be defined recursively by

P0(x) = 1

P1(x) = x

Pn(x) =1

n[(2n− 1)xPn−1(x)− (n− 1)Pn−1(x)] .

Legendre polynomials are standardized by Pn(1) = 1, and are once again only orthog-onal, as their norm hn is given by,

hn =2

2n+ 1.

7.3 Gamma Function

There are a number of different definitions of the gamma function [1], however we willstate what is applicable in our context.

Using, what is named Euler’s Integral, the gamma function may be defined as

Γ(x) =

∫ ∞0

tz−1e−tdt, Re z > 0

25

A number of noteworthy identities of the gamma function include

Γ(x)Γ(1− x) =π

sinπx,

Γ(1

2) =√π,

and for integer values of n,

Γ(n+ 1) = 1 · 2 · 3 · ... · (n− 1) · n = n!

7.4 Lp Functions

The set of Lp function on the interval a ≤ x ≤ b [11] are defined as all functionsthat satisfy

Lp(a, b) ={f∣∣∣ ∫ b

a

|f(x)|pdx <∞}.

Here the Lp norm of a function f is denoted

||f || =(∫ b

a

|f(x)|pdx) 1p

.

26

References

[1] M.Abramowitz and I.Stegun, Handbook of Mathematical Functions, National Bu-reau of Standards, (1964), pp 255-257, 773-776.

[2] J.W Brown and R.V. Churchill, Fourier Series and Boundary Value Problems.Fifth edition, McGraw-Hill, INC, (1993).

[3] A. Gelb and S. Gottlieb, The Resolution of the Gibbs Phenomenon for FourierSpectral Methods,, (2006).

[4] D. Gottlieb and C. Shu, On the Gibbs Phenomenon and its Resolution, SIAMReview,Vol.39, (1997), pp.644-668.

[5] D. Gottlieb, C. Shu, A. Solomonoff and H. Vandeven, On the Gibbs phenomenonI: recovering exponential accuracy from the Fourier partial sum of a non periodicanalytic function, Journal of Computational and Applied Mathematics Vol. 43,(1992), pp 81-92.

[6] D. Gottlieb, C. Shu, On the Gibbs Phenomenon III: Recovering exponential accu-racy in a subinterval from a spectral partial sum of a piecewise analytic function,SIAM Journals on Numerical Analysis Vol. 33, (1996), pp 280-290.

[7] D. Gottlieb, C. Shu, On the Gibbs Phenomenon IV: Recovering exponential ac-curacy in a sub-interval form a Gegenbauer partial sum of a piecewise analyticfunction, Mathematics of Computation Vol. 64, (1995), pp 1081-1095.

[8] D. Gottlieb, C. Shu, On the Gibbs Phenomenon V: Recovering exponential ac-curacy from collocation point values of a piecewise analytic function, NumerischeMathematic Vol.71, (1995), pp.511-526.

[9] G. James, Modern Engineering Mathematics, Adison-Wesley, (1993), pp 756-759.

[10] C. Karlsson and P. Kolstrom, Gibbs fenomen, Vaxjo University, (2000).

[11] E. Kreyszig, Introductory Functional Analysis with Applications, John Wiley andSons,(1978), pp 62-63.

[12] E. Weisstein, Orthogonal Polynomials, From MathWorld, A WolframWeb Resource, http://mathworld.wolfram.com/OrthogonalPolynomials.html,(2015/05/26).

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