Non-Commutative Harmonic Analysis on

Compact Groups

Alexander Hiller

Supervised by Professor Anthony DooleyUniversity of Technology Sydney

Vacation Research Scholarships are funded jointly by the Department of Education and

Training and the Australian Mathematical Sciences Institute.

Contents

1 Prelude 4

1.1 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Statement of Authorship . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Introduction 4

3 Group Theory 6

3.1 Definition of a Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3.2 Finite Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3.3 Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3.4 Cosets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3.5 Normal Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3.6 A Brief Taxonomy of Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.6.1 Abelian Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.6.2 Compact Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.6.3 Locally Compact Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.6.4 Special Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.6.5 SU(2) in General Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.6.6 SO(3) in General Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.7 Matrix Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.7.1 Relationship Between SU(2) and SO(3) . . . . . . . . . . . . . . . . . . . . . . 8

4 Measure Theory 9

4.1 Lebesgue Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4.2 Haar Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4.2.1 Proof of Haar Measure on (R,+) . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4.2.2 Proof of Haar Measure on (R\0,×) . . . . . . . . . . . . . . . . . . . . . . . . . 10

4.3 Other Haar Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4.3.1 Haar’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

5 Representation Theory 12

1

5.1 What is Representation Theory? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

5.2 Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5.3 Irreducible Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5.4 The Peter-Weyl Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

5.4.1 Prelude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

5.4.2 The Actual Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

5.5 Representation of SU(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5.6 Representation of SO(3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

6 Harmonic Analysis on Locally Compact Abelian Groups 16

6.1 Schur’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

6.2 Dual Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

6.3 Pontryagin Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

7 Harmonic Analysis on Compact Groups 17

7.1 Harmonic Analysis on Finite Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

7.2 Harmonic Analysis on Compact Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . 18

7.3 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

7.4 Plancherel Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

7.4.1 Finite Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

7.4.2 Compact Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

7.5 Orthogonality Relations for Compact Groups . . . . . . . . . . . . . . . . . . . . . . . 19

7.5.1 Schur’s Orthogonality Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

8 Next Steps 19

8.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

8.2 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

9 Conclusion 20

10 Appendix 22

10.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

10.2 Code for Producing Matrix Elements of SU(2) Representation . . . . . . . . . . . . . . 22

10.3 Base Code for Finite Group Fourier Transform . . . . . . . . . . . . . . . . . . . . . . 23

2

11 Background 25

11.1 Arriving at the Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

11.2 The Fourier Transform and its Properties . . . . . . . . . . . . . . . . . . . . . . . . . 26

11.3 General Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

11.3.1 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

11.3.2 Translation Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

11.3.3 Parseval-Plancherel Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

11.3.4 Orthogonality Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

11.4 Some Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

11.4.1 Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

11.4.2 Orthonormal Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

11.4.3 Trace Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

11.4.4 Groups of Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

11.5 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

11.5.1 Duality in Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

11.5.2 Duality in Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

11.6 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

11.6.1 Direct Sum (⊕) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

11.6.2 Tensor Product (⊗) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

11.6.3 Permutation Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

11.6.4 Dense Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3

1 Prelude

1.1 Acknowledgements

I would like to thank Anthony Dooley for his mentorship both before and throughout this project, itis something I have come to feel incredibly lucky to have.

Peter McLean, thank you for letting me sit in your lab over this last summer (and beyond). Youallowed me to talk incessantly about the ins-and-outs of mathematics, and beared with me whilst Itell you how I figured something out, only to later listen to me say how I had it all wrong.

And to my family, who have shown me their endless and endless and unwaivering support.

1.2 Abstract

We elucidate the established field of abstract harmonic analysis, building up the theory from anunderstanding of the Fourier transform. The progression starts with a derivation of the Fouriertransform, details of its properties, some miscellaneous notes on linear algebra and other miscellaneouspieces of mathematics.

From there we build up some knowledge of group theory, measure theory (mainly the Haar measure)and representation theory. We then present some facts about harmonics analysis on locally compactAbelian groups and then move to the results of harmonic analysis on compact groups (finite, SU(2),and SO(3)).

After this, we propose some next steps in terms of theory and application.

1.3 Statement of Authorship

We have sourced the information present from many texts, though the notation and structure defintelyvary from any of those listed in the references.

The theoretical knowledge here is well established and no new theorems are presented, but this projecthas been a chance to understand the process of learning graduate-level mathematics, especially forsomeone in the engineering discipline.

Auxiliary software is original and was written to produce representations and to build a base that maybe used to perform transforms on finite groups. This can be found in full in the appendix.

2 Introduction

Joseph Fourier in his attempt to understand heat conduction discovered the phenomenon that periodicfunctions can be represented by an infinite sum of weighted sinusoids

f(x) =

∞∑n=0

an cos (nx) + bn sin (nx) .

4

Of course, sin (x) and cos (x) have periodicity of 2π, but we can rewrite the function with an arbitrary(but finite) period (1/ξ0) by some algebraic massaging

∞∑n=0

an cos (nx) + bn sin (nx) −→∞∑n=0

an cos (2πξ0nx) + bn sin (2πξ0nx) .

Fourier series can also be expressed using the orthogonality of√−1 and the real numbers, exploiting

Euler’s identity1

∞∑n=0

an cos (nx) + bn sin (nx) ∼=∞∑

n=−∞Gne

2πinξ0x.

The coefficients (Gn) of the Fourier series are also calculable, namely by the formula

Gn =1

ξ0

∫ 1/(2ξ0)

−1/(2ξ0)f(x)e−2πinξ0x dx

for some function f : R −→ C with period 1/ξ0. These facts have been well known for hundreds ofyears and are well exposited in any good education in the quantitative sciences [McLean, 2015].

The insight of Fourier series has been generalised to non-periodic functions, namely through the Fouriertransform, which we will derive in the next section. This has allowed mathematicians, engineers andscientists to both break-down complex functions (Fig 1, right) into simpler constituents as well asuse Fourier series to approximate functions that are not realisable with real-world components (Fig 1,left).

But the functions f : R −→ C are quite a small amount of those that exist. What if we could takefunctions from something more general? Like that of a group from abstract algebra? This immediatelyraises questions about what functions and groups this could be done for and whether the propertiesof these transformations still exist or have some analogue.

We hope to answer some of these questions and illuminate a field of study rich with intersections ofdifferent fields of mathematics.

Figure 1: (Left) Using Fourier series, we can generate the square-wave function. (Right) Decompositionof functions into constituents using the Fourier transform.

1Euler’s identity is eiθ = cos (θ) + i sin (θ).

5

3 Group Theory

All subsections in the Group Theory section take a lot from Artin’s Algebra. Most of what is hereis exposited by Artin in a much more gentle and intuitive fashion. We point those with a desire forgreater understanding to his text [Artin, 2010].

3.1 Definition of a Group

A group, G requires a set of elements (X), an associative law (◦) and some structure amongst the set.By structure what is meant is that there must be an identity element

∃ e ∈ G | g ◦ e = e ◦ g = g, ∀g ∈ G,

inverse elements∃ g−1 ∈ G | g−1 ◦ g = e,∀g ∈ G,

and closure(g ◦ h) ∈ G | ∀g, h ∈ G.

3.2 Finite Groups

A finite group G is one who has a finite number of elements. We say |G| <∞ if G is a finite group.

3.3 Subgroups

H is a subgroup of G if it shares the group action and some subset of elements from G. H also has tosatisfy the structural attributes of a group (identity, inverses and closure) to be a subgroup.

If H is a subgroup of G we write:H < G

3.4 Cosets

A coset of a subgroup H is a set of elements formed by taking some g ∈ G and computing

g ◦H = {g ◦ h | h ∈ H}

where ◦ is the group action of H and G.

3.5 Normal Subgroups

A normal subgroup is one that forms identical left and right coset spaces. Explicitly we can state thisas

g ◦H = H ◦ g | ∀g ∈ G

or more commonlyg ◦H ◦ g−1 = H.

6

If H is a normal subgroup of G we writeH / G.

It is helpful to note all subgroups with a commutative associative law (termed Abelian groups) willbe normal. It’s easy to see why with a one-line proof

g ◦H ◦ g−1 = H ◦ (g ◦ g−1) = H ◦ e = H.

3.6 A Brief Taxonomy of Groups

3.6.1 Abelian Groups

A group with a commutative law, i.e. a ◦ b = b ◦ a. e.g. (R,+), (R\0,×) and (C\0,×) [Artin, 2010].

3.6.2 Compact Groups

Will be thought of as closed and bounded to save ourselves an exposition in topology [Munkres, 2000].Colloquially, this is the topological way of saying finite. All finite groups, and the Lie groups SU(2)and SO(3) are compact.

For example, through the method of Mobius transformations SU(2) can be shown to be isomorphicto a four-dimensional unit-sphere S3 [Chirikjian and Kyatkin, 2001]. This sphere is finite in volume,meaning it does not cover all of four-dimensional space. Because of this (and other proofs) we canclassify SU(2) as compact.

3.6.3 Locally Compact Groups

We can think of closed and bounded in a neighbourhood of a point, e.g. (R,+) is locally compact butnot compact [Munkres, 2000].

For example, the real line is an infinitely long, and so is not compact. However, any proper subset ofthe real line is locally compact, as it would have to be finite.

3.6.4 Special Matrices

Special groups of matrices have determinant = +1. Some of the groups (Lie groups, in fact) underexamination for harmonic analysis are: SU(2), the special group of unitary matrices and SO(3), thespecial group of orthogonal matrices.

3.6.5 SU(2) in General Form

For the uninitiated, the general form of SU(2) is stated as:

SU(2) :=

{[α β

−β α

]: α, β ∈ C, αα+ ββ = 1

}

7

And indeed it is unitary [α β

−β α

] [α β

−β α

]∗=

[α β

−β α

] [α −ββ α

]=

[αα+ ββ 0

0 ββ + αα

]=

[1 00 1

]

3.6.6 SO(3) in General Form

Every matrix in SO(3) can be written as a multiplication of three matricescos (ϕ1) − sin (ϕ1) 0sin (ϕ1) cos (ϕ1) 0

0 0 1

1 0 00 cos (θ) − sin (θ)0 sin (θ) cos (θ)

cos (ϕ2) − sin (ϕ2) 0sin (ϕ2) cos (ϕ2) 0

0 0 1

where ϕ1, ϕ2 and θ are Euler angles and have range of ϕ1,2 ∈ [0, 2π), and θ ∈ [0, π].

3.7 Matrix Lie Groups

First explored by Sophus Lie2, all matrix Lie groups are subgroups of GL(n;C). They share theproperty that for a sequence of matrices Am that converge to a limit matrix A, the limit is also in thegroup. Hall discusses this property, examples and counter examples at length in his first chapter [Hall,2015].

The unitary group of matrices U(n;C) and the orthogonal group of matrices O(n;C) are subgroupsof GL(n;C), contain their limit points and the identity matrix, making them Lie groups as well.

Lie groups also have manifold properties, though they won’t be discussed as these properties are notthe focus of this report.

3.7.1 Relationship Between SU(2) and SO(3)

SU(2) is related to SO(3) by a surjective mapping, that is SU(2) can be mapped onto SO(3).

τ : SU(2) −→ SO(3)

Chirikjian proves this using Mobius transformations to show that SU(2) is isomorphic to the 3-sphere (a four-dimensional sphere) S3 and that SO(3) is isomorphic to the three-sphere’s upperhalf [Chirikjian and Kyatkin, 2001], but we state the results here without proof.

SU(2) ∼= S3

SO(3) ∼= S3≥0

So we assert that ∃ τ ′ such thatτ ′ : S3 −→ S3

≥0

2Pronounced Lee.

8

implying that ∃ τ such thatτ : SU(2) −→ SO(3).

In fact, we can also express SO(3) as a quotient group [Hewitt and Ross, 1965]

SO(3) = SU(2)/{I,−I}.

4 Measure Theory

A measure is a function on a set that maps it to the real line

µ : S −→ R.

This allows for a scalar value to be placed on sets. This body of theory is paramount in formulatingthe mathematical study of analysis.

Say our set S was all points x ∈ R3 with |x| ≤ 1. Then a measure µ(S) could be interpretted as avolume of a sphere. Similarly, if our points lied in R2 and again satisfied |x| ≤ 1 then we would havea measure µ(S) that could be interpretted as an area of a circle.

4.1 Lebesgue Measure

For the sake of brevity, we can say that a Lebesgue measure is one that obtains the length of a giveninterval.

Example:µ([1, 2]) = (2− 1) = 1

−3 −2 −1 0 1 2 3

There is a lot to say on the Lebesgue measure, though that is not the focus of this report, this topicis well discussed by Tao [Tao, 2016]. The very brief introduction here is only in service to themeasure we will be using for harmonic analysis, the Haar measure.

4.2 Haar Measure

The Haar measure is invariant under translation or better said: invariant under group action

µL(S) = µL(g ◦ S) | G := (X, ◦), g ∈ G, S ⊂ G.

Allows us to measure the size/length/area/volume of a subset of a group. This then allows us tointegrate over a group, i.e.

µL(S) =

∫SdµL(g)

where S ⊂ G and µL is the left Haar measure [Folland, 2015].

9

The Haar measure is often referred to as the left Haar measure because many of the groups that it isused on are non-Abelian. We must specify whether the measure is invariant to group action on theleft (left Haar measure) or invariant to group action on the right (right Haar measure). Groups whohave identical left and right Haar measures are called unimodular.

Let us now detail some simple derivations of Haar measures on (R,+) and (R\0,×).

4.2.1 Proof of Haar Measure on (R,+)

Assertion: The Haar measure on the group (R,+) is simply the Lebesgue measure.

Proof :

S ⊂ RS′ := {c ◦ s | s ∈ S, c = constant in R}S = [a, b]

S′ = [c+ a, c+ b]

Assume dµL(x) = dx, i.e. that the left Haar measure is the Lebesgue measure (for those unfamiliarwith the Lebesgue measure, this can be simply thought of as the dx we all know and love).

µL(S) =

∫ b

adµL(x) =

∫ b

adx = (b− a)

µL(S′) =

∫ c+b

c+adµL(x) =

∫ c+b

c+adx = (c+ b)− (c+ a) = (b− a)

µL(S) = µL(S′) and so the Haar Measure on (R,+) is the Lebesgue Measure

dµL(x) = d(x) = dx.

4.2.2 Proof of Haar Measure on (R\0,×)

Assertion: The Haar Measure on (R\0,×) is dx|x|

Proof :

S ⊂ RS′ := {c ◦ s | s ∈ S, c = constant in R\0}S = [a, b]

S′ = [ca, cb]

Assuming dµL(x) = dx doesn’t work!

µL(S) =

∫ b

adµL(x) =

∫ b

adx = (b− a)

µL(S′) =

∫ cb

cadµL(x) =

∫ cb

cadx = (cb)− (ca) = c(b− a)

and soµL(S) 6= µL(S′).

10

We have to do a normalisation, meaning we assume dµL(x) = dx/|x|.

µ(S) =

∫ b

adµL(x) =

∫ b

a

dx

|x|= ln(|b|)− ln(|a|) = ln

(∣∣∣∣ ba∣∣∣∣)

µ(S′) =

∫ cb

cadµL(x) =

∫ cb

ca

dx

|x|= ln(cb)− ln(ca) = ln

(∣∣∣∣ cbca∣∣∣∣)

= ln

(∣∣∣∣ ba∣∣∣∣)

Therefore µL(S) = µL(S′).

4.3 Other Haar Measures

Tabulations of these measures can be found in most books on Abstract Harmonic Analysis, as inFolland [Folland, 2015].

Group (X, ◦) Left Haar Measure (µL)

(R,+) dx

(R\0,×) dx/|x|(C\0,×) dx dy/(x2 + y2)

(GLn,×) dX/|det (X) |n(H\0,×) dx dy dz dw/(x2 + y2 + z2 + w2)2

4.3.1 Haar’s Theorem

There exists µL that is unique up to a multiplicative constant for any locally compact group. It isalso:

1. Countably additive.

2. Non-trivial on the Borel (measurable) sets of G.

3. Left-translation invariant µ(g ◦ S) = µ(S) | S ⊂ G.

4. Finite on compact sets3.

5. Algebraic groups for whom the left and right Haar measure are the same are calledunimodular [Bump, 2013].

The measure was formulated by Haar in 1935, then Cartan and Weil proved the uniqueness andexistence theorems in 1940 [Folland, 2015]. So what does this mean? We can now integrate over

3Here we are translating the idea of finite from topology (compactness) to finite values of the real line. This is abeautiful connection between topology and analysis that is provided by the Haar measure.

11

groups

µL(G) =

∫GdµL(g) | g ∈ G

µL(SO(3)) =

∫SO(3)

dµL(g) | g ∈ SO(3)

µL(SU(2)) =

∫SU(2)

dµL(g) | g ∈ SU(2).

Note that often the group’s measure is normalised, meaning the Haar measure of the entire group is+1, e.g.

µL(G) = 1

µL(SO(3)) = 1

µL(SU(2)) = 1.

5 Representation Theory

5.1 What is Representation Theory?

Representation Theory is a way of mapping groups to vector spaces. Once there, we can use thetools of linear algebra to examine algebraic groups [Artin, 2010]

ρ : G −→ V.

Note that ρ is used to indicate a representation and σ to indicate an irreducible representation.

Caution:Two representations can be equivalent but vary in their basis, or be composed of several smallerrepresentations (more on that in section 5.3) [Artin, 2010].

Example:We can take some elements of the symmetry group S3 and write them in permutation notation

e = (1) x = (1 2 3) y = (1 2).

We can also express this group as

S3 = {e, x, x2, y, xy, x2y}.

If we consider ρ to be the map from the group to a permutation matrix,

ρ : S3 −→ V

then we can write this representation as

ρ(S3) =

{[1 0 00 1 00 0 1

],

[0 0 11 0 00 1 0

],

[0 1 00 0 11 0 0

],

[0 1 01 0 00 0 1

],

[0 0 10 1 01 0 0

],

[1 0 00 0 10 1 0

]}.

We now have a set of matrices that obey the multiplication table of the group and each representone group element.

12

5.2 Characters

A character is a function from a group to a field (commonly C). For our purposes, all charactersmentioned will take the trace of a representation

χ : ρ(g) 7−→ trace(ρ(g)).

Characters simplify representation theory because of some properties of the trace operator:

1. trace(A) is independent of the basis of A.

2. The trace of a representation of a group element g is constant over a conjugacy class of G.

From now on, we will write characters as trace(ρ(g)) or χ(g). To elaborate on conjugacy classes, it’sbest to give an example.

Example:If we define the group elements in permutation notation, e = (1), x = (1 2 3) and y = (1 2). ThenS3 = {e, x, x2, y, xy, x2y} and its conjugacy classes are:

conj(S3) = {{e}, {x, x2}, {y, xy, x2y}}

so we would have the equalitiesχ(x) = χ(x2)

χ(y) = χ(xy) = χ(x2y).

Note: An element of a group can belong to one and only one conjugacy class.

With groups being partitioned by their conjugacy classes, characters give us a tool to use onrepresentations that removes distractions generated by the machinery of the vector space (namely,the basis).

5.3 Irreducible Representations

For finite groups, Maschke’s Theorem tells us that every representation of a finite group G is a directsum of irreducible representations.

ρ ≈ n1ρ1 ⊕ . . .⊕ nrρrρ1, . . . , ρr are the irreducible representations of G, where

niρi = ρi ⊕ . . .⊕ ρi︸ ︷︷ ︸ni copies of ρi

.

An irreducible representation is one that cannot be made up of smaller representations. We willdenote σj to be the j-th irreducible representation4 of G [Artin, 2010].

Example:The irreducible unitary representations of the S3 group (the symmetries of a triangle) are σA, σΣ andσT .

4As opposed to ρj just being some representation of G, that may or may not be irreducible.

13

σA gives a two dimensional representation of the triangle, mimicking its 2π/3 roations with thetransformations provided by using σA(x), similarly flipping along one of its axes can be done byusing σA(y). σΣ is the sign representation, it can be calculated by taking the determinant of thepermutation that represents it, and σT is the trivial representation that takes all elements of thegroup to the trivial (1× 1) vector of +1.

Putting S3 into each irreducible representation we get:

σA(e) =

[1 00 1

]σA(x) =

[cos(

2σ3

)− sin

(2σ3

)sin(

2σ3

)cos(

2σ3

) ] σA(y) =

[1 00 −1

]σΣ(e) = [1] σΣ(x) = [1] σΣ(y) = [−1]

σT (e) = [1] σT (x) = [1] σT (y) = [1]

We have given the representations only for e, x and y because the other three elements of S3 can bedetermined from the values given.

5.4 The Peter-Weyl Theorem

5.4.1 Prelude

The Peter-Weyl theorem makes use of the notion of an Lp space, particularly the L2 space. An L2

space is one that consists of all square integrable functions over a domain D. We can state this moreexplicitly as ∫

D|f(x)|2 dx <∞

In terms of locally compact groups, we will be more closely examining the space of L2(G), which isthe space of square integrable functions over the group G.

5.4.2 The Actual Theorem

We define EA to generally be the span of the columns of matrix A, [σ] ∈ G to be the irreducibleunitary representations of G, the dual object of G, and C(G) to be the continuous functions on thegroup G. However we define E to be

E := the linear span of⋃

[σ]∈G

Eσ

i.e. the combined span of all the [σ] ∈ G. Let G be a compact group. Then

• E is uniformly dense in C(G)

• L2(G) =⊕

[σ]∈G Eσ

• and if σij is given by

{√dσσij : i, j = 1, . . . , dσ, [σ] ∈ G}

then [σ] ∈ G is an orthonormal basis for L2(G).

14

Consequently, each [σ] ∈ G occurs with multiplicity dσ [Folland, 2015].

This establishes an L2 isomorphism between compact groups (SU(2) and SO(3)) and their unitaryirreducible representations. This is comparable to the isomorphism that exists between the regularFourier transform stated via Plancherel’s theorem in section 11.3.3.

We use the Peter-Weyl theorem as a cornerstone for establishing an L2 theory for compact Liegroups which leads to establishing their equations of harmonic analysis.

5.5 Representation of SU(2)

For SU(2), a compact group, we have the Peter-Weyl theorem as an analogue for the Maschke’sTheorem. Note that because SU(2) is non-Abelian, its representations are in dimensions greaterthan one (a consequence of Schur’s lemma) [Folland, 2015].

In fact, from the mid-20th century two-volume tome of Hewitt and Ross, we can source the followingformula for generating the matrix coefficients for representations of SU(2)

σ(l)j,k(u) = (−1)j−k

((l − j)!(l + j)!

(l − k)!(l + k)!

)1/2

×

min(l+k,l−j)∑s=max(0,k−l)

(−1)s(l + ks

)(l − k

l − j − s

)αl−j−s α l+k−s βs β

j−k+s.

Regarding notation, here j and k are indexing variables, l is a half integer increment (i.e.l = {0, 1

2 , 1,32 . . .}), and this produces an entry of an (m×m) matrix, where m = 2l + 1 [Hewitt and

Ross, 1965] [Gou, 2019]. The parantheses around vertical entries indicate the binomical coefficient5,e.g. (

l + ks

)=

(l + k)!

s!(l + k − s)!.

To calculate the representations by hand is a laborious process, so we have used a computer to dothem using original code provided in appendix section 10.2.

l = 0 ⇒ m = 2(0) + 1 = 1 ⇒ (1× 1)[1]

l =1

2⇒ m = 2

(1

2

)+ 1 = 2 ⇒ (2× 2)[

α β

−β α

]

l = 1 ⇒ m = 2(1) + 1 = 3 ⇒ (3× 3)

5Sometimes called n-choose-k.

15

α2√

2αβ β2

−√

2αβ αα− ββ√

2βα

β2 −

√2αβ α2

l =

3

2⇒ m = 2

(3

2

)+ 1 = 4 ⇒ (4× 4)

α3√

3α2β√

3αβ2 β3

−√

3α2β −2αββ + α2α 2αβα − β2β√

3β2α√3αβ

2 −2ααβ + ββ2

αα2 − 2βαβ√

3βα2

−β3 √3αβ

2 −√

3α2β α3

l = 2 ⇒ m = 2(2) + 1 = 5 ⇒ (5× 5)

α4 2α3β√

6α2β2 2αβ3 β4

−2α3β α3α− 3α2ββ −0.5√

6(−2α2βα+ 2αβ2β

)3αβ2α− β3β 2β3α

√6α2β

2 −√

6(

3α2αβ−3αββ2)

3 α2α2 − 4αβαβ + β2β2 −1

3

√6(−3αβα2 + 3β2αβ

) √6β2α2

−2αβ3

3ααβ2 − ββ3 −

√6(

2αα2β−2βαβ2)

2 αα3 − 3βα2β 2βα3

β4 −2αβ

3 √6α2β

2 −2α3β α4

l =5

2⇒ m = 2

(5

2

)+ 1 = 6 ⇒ (6× 6)

α5 √5α4β

√10α3β2 √

10α2β3 √5αβ4 β5

−√5α4β −4α3ββ + α4α −3

√2α2β2β + 2

√2α3βα −2

√2αβ3β + 3

√2α2β2α 4αβ3α − β4β

√5β4α√

10α3β2

3√2α2ββ

2 − 2√2α3αβ 3αβ2β

2 − 6α2βαβ + α3α2 −6αβ2αβ + 3α2βα2 + β3β2

3√2αβ2α2 − 2

√2β3αβ

√10β3α2

−√10α2β

3 −2√2αββ

3+ 3√2α2αβ

26αβαβ

2 − 3α2α2β − β2β3 −6αβα2β + α2α3 + 3β2αβ

22√2αβα3 − 3

√2β2α2β

√10β2α3

√5αβ

4 −4ααβ3+ ββ

43√2αα2β

2 − 2√2βαβ

3 −2√2αα3β + 3

√2βα2β

2αα4 − 4βα3β

√5βα4

−β5 √5αβ

4 −√10α2β

3 √10α3β

2 −√5α4β α5

5.6 Representation of SO(3)

Because of the two-to-one surjective homomorphism from SU(2) to SO(3) mentioned in section3.7.1, the representations of SO(3) correspond to the non-negative integer values of l mentioned inthe representation of SU(2) [Hewitt and Ross, 1965].

6 Harmonic Analysis on Locally Compact Abelian Groups

This section on locally compact Abelian groups is brief and only to elucidate the importance ofduality in harmonic analysis and provide parallels when we examing the non-Abelian case.

6.1 Schur’s Lemma

Schur’s lemma is two-parted and we quote it here from Folland [Folland, 2015] without proof.

16

1. A unitary representation σ of G is irreducible if and only if C(σ) contains only scalar multiplesof the identity.

2. Suppose σ1 and σ2 are irreducible unitary representations of G. If σ1 and σ2 are equivalentthen C(π1, π2) is one dimensional; otherwise, C(π1, π2) = {0}.

A result of Schur’s lemma is that if G is Abelian, then every irreducible representation of G isone-dimensional.

6.2 Dual Groups

Dual groups (G) are the set of irreducible unitary representations (IURs) of a group, G. As aconsequence of Schur’s lemma, all Abelian groups have IURs that are one dimensional, meaning theylie in Hilbert space we can represent with C (recall that because of the unitary property of theserepresentations, each dimension is complex).

For locally compact Abelian groups, these IURs are often termed the characters of the group.

We now list some dual groups.

R ∼= RT ∼= ZZ ∼= T

Theorem: If G is discrete, then G is compact. If G is compact then G is discrete [Folland, 2015].

6.3 Pontryagin Duality

Pontryagin6 duality theorem states that the characters of the characters is isomorphic to the groupitself!

G ∼= G

[Folland, 2015].

Caution: The Pontryagin duality theorem states that G −→ G is an isomorphism between

topological groups, not that they are equivalent. This fact is often overlooked and isomorphisms areoften stated as equivalences.

7 Harmonic Analysis on Compact Groups

We now crescendo and state the results of harmonic analysis on both finite groups and compact Liegroups.

6Sometimes spelt Pontrjagin.

17

7.1 Harmonic Analysis on Finite Groups

f(σj) =∑g∈G

f(g)σj(g−1)

Where f(σj) is the harmonic analysis of f(g) at σj .

If we want the entire spectrum, we will have to do it for each irreducible unitary representation.

F−1(f) = f(g) =1

|G|

α∑j=1

dj trace(f(σj)σj(g))

As shown by the formula, we will need all transforms of all irreducible unitary representations (i.e.the whole spectrum of f) {σ1, σ2, . . . , σα} to recover the function [Chirikjian and Kyatkin, 2001].

7.2 Harmonic Analysis on Compact Lie Groups

f(σi) =

∫Gf(g)σi(g

−1) dµL(g)

F−1(f)

= f(g) =∑σi∈G

dσi trace(f(σi)σi(g)

)

G is the collection of all possible IURs, σ, and is countably infinite. The set of all transforms{f(σi)

}∞i=1

is termed the spectrum of f [Chirikjian and Kyatkin, 2001].

7.3 Convolution

Citing the results provided by Chirikjian which holds for both finite groups and compact Lie groups,we have

F(f1 ∗ f2)(σj) = f2(σj)f1(σj)

where the result is not commutative [Chirikjian and Kyatkin, 2001].

7.4 Plancherel Theorem

The main differences in the Plancherel theorem for finite and compact groups is that compact Liegroups have a countably-infinite number of representations. Constrastingly, finite groups will haveonly have a finite number, here denoted by α.

7.4.1 Finite Groups

1

|G|∑g∈G|f(g)|2 =

α∑j=1

dσj trace(f(σj)

∗f(σj))

[Terras, 2001]

18

7.4.2 Compact Lie Groups

∫G|f(g)|2 dµL(g) =

∞∑i=1

dσitrace(f(σi)

∗ f(σi))

[Folland, 2015]

7.5 Orthogonality Relations for Compact Groups

Recall that for the harmonics of a complex-exponential Fourier series, each is orthogonal from oneanother by the inner product ∫ 2π

0einxeimx dx =

{2π n = m

0 n 6= m.

We now have the IURs filling the place of einx for a Fourier series and so now, our IURs should beorthogonal.

In fact, this is the case, as shown by the Schur orthogonality relations, a result that follows fromSchur’s lemma.

7.5.1 Schur’s Orthogonality Relations

Let σ and σ′ be IURs of G and consider Eσ and Eσ′ to be subspaces of L2(G), then

1. Then if σ 6= σ′ then Eσ ⊥ Eσ′

2. If {ej} is any orthonormal basis for Hσ and σij is given by

σij(g) = 〈σ(g)ej , ei〉

then {√dσ σij : i, j = 1, . . . , dσ} is an orthonormal basis for Eσ [Folland, 2015].

We can also state this in a manner similar to our original harmonics∫G〈σ(g)ej , ei〉

⟨σ′(g)ej′ , ei′

⟩dµL(g) =

{δijδi′j′

1dσ

σ = σ′

0 σ 6= σ′

where δij is the Kronecker-delta function defined by

δij =

{1 i = j

0 i 6= j.

8 Next Steps

We can split the next steps into theory and application.

19

8.1 Theory

Next steps would be to understand the techniques of harmonic analysis of non-compact groups, likeSE(3), the special Euclidean three-dimensional group7. SE(3) is the collection of all rotations andtranslations in R3 that do not cause reflections.

These kinds of movements are the ones we experience in the mechanics of everyday life and so SE(3)has myriads of uses in application. The complexity lies in the non-compactness of the group, forwhich the theory is spotty in places and considerably more complex than for compact groups.

8.2 Application

The next steps in terms of application is two-parted:

1. Development of a library for the finite group Fourier transform. Given input of only a list ofelements, a group action and function to be analysed, it would validate these inputs and returna Fourier transform. The beginnings of this can be seen in appendix section 10.3.

2. Development of a fast Fourier transform (FFT) library to perform harmonic analysis onfunctions for SE(3). This may then be used to analyse orbital mechanics, robot kinematics andnoise-reduction in sensors. This software would build on the work detailed by Chirikjian andKyatkin [Chirikjian and Kyatkin, 2001]. The finished library should function on embeddedsystems, whilst being time and resource sensitive. The candidate language is currently C++ 11.

9 Conclusion

This work has been heavily focused on building a bridge between traditional Fourier analysis,through the finite group Fourier transform and to the harmonic analysis of compact Lie groups.

Current research involves the theory of non-compact groups, though understanding the currentstate-of-the-art involves first learning the theory of compact groups, which is one of the outcomesthis project intended to accomplish.

7Some authors refer to this group as M(3).

20

References

[Artin, 2010] Artin, M. (2010). Algebra. Pearson, Boston, second edition.

[Bump, 2013] Bump, D. (2013). Lie Groups. Springer, New York, USA, second edition.

[Chirikjian and Kyatkin, 2001] Chirikjian, G. and Kyatkin, A. (2001). Engineering Applications ofNoncommutative Harmonic Analysis. CRC Press, London.

[Folland, 2015] Folland, G. (2015). A Course in Abstract Harmonic Analysis. CRC Press, BocaRaton, FL, USA, second edition.

[Gou, 2019] Gou, T. (2019). Representations of compact lie groups. Master’s thesis, UniversityTechnology Sydney, Sydney, NSW, Australia.

[Hall, 2015] Hall, B. (2015). Lie Groups, Lie Algebra, and Representations. Springer, Notre Dame,IN, USA, second edition.

[Hewitt and Ross, 1965] Hewitt, E. and Ross, K. (1965). Abstract Harmonic Analysis Volume 2.Springer-Verlag, Berlin, Germany, first edition.

[McLean, 2015] McLean, P. (2015). Signals and systems. Lecture Notes at University of TechnologySydney, Sydney, NSW, Australia.

[Munkres, 2000] Munkres, J. (2000). Topology. Pearson, New Jersey, USA, second edition.

[Neuenschwander, 2015] Neuenschwander, D. E. (2015). Tensor Calculus for Physics, A ConciseGuide. John Hopkins University Press, Baltimore, MD, USA, first edition.

[Osgood, 2019] Osgood, B. (2019). The fourier transform and its applications. Lecture Notes atStanford University, Stanford, CA, USA.

[Strang, 2006] Strang, G. (2006). Linear Algebra and Its Applications. Thomson, Brooks/Cole,Boston, MA, USA, fourth edition.

[Tao, 2016] Tao, T. (2016). Analysis II. Springer, Los Angeles, California, USA, third edition.

[Terras, 2001] Terras, A. (2001). Fourier Analyis on Finite Groups and Applications. CambridgeUniversity Press, San Diego, California, USA, first edition.

21

10 Appendix

We have notation, background theory and code in the appendix.

Software is kept in a private github repository and can be made available for anyone wishing to forkthe repository, though there will likely have been changes after the publication of this report.

The author can be found for software requests at github.com/xandhiller.

10.1 Notation

Notation Meaning

⇒ Logically leads to

∴ Therefore

∵ Because

⊂ Subset, i.e. if U ⊂ X then all elements of U can be found in X.

⊆ Proper subset, i.e. U ⊂ X and U 6= X.

∼ Similar or related by an equivalence relation

' Homomorphic∼= Isomorphic

∈ Contained in

∃ Exists

N Set of natural numbers excluding zero, e.g. 1, 2, 3, . . .

Z Set of integers, e.g. −2, −1, 0, 1, 2, . . .

Q Set of rational numbers, i.e. those that can be expressed as a ratio of two integers.

R Set of real numbers, e.g. 0, 1, 2, e, 3.01, π, 4, . . .

C Set of complex numbers, e.g. a+ ib where i =√−1 and a, b ∈ R.

X −→ Y Set X is mapped to the set Y .

x 7−→ x2 The point x is mapped to the point x2.

conj(G) The conjugacy classes of the group G.

|X| The cardinality of the set X, i.e. how many elements X has.

|G| The cardinality of the group G, i.e. how many elements G has.

ρ, ρg or ρ(g) Some representation of the group element g. Not necessarily irreducible.

σ, σg or σ(g) Some irreducible representation of the group element g.

Caution:Caligraphic text such as SO(n), SU(n) or GL(n) is used to denote Lie groups. Should the need ariseto denote collections (a set of sets) then we will also use caligraphic text, though the differencebetween a Lie group and a collection should be clear from context.

10.2 Code for Producing Matrix Elements of SU(2) Representation

1 #! / usr /bin /python3 . 7 −−23 from sympy import Symbol , Matrix , i n i t p r i n t i n g , f a c t o r i a l , binomial , s q r t4 from sympy import conjugate as conj5 from numpy import arange , power6 from sympy . matr i ces import z e ro s7 from sympy import l a t ex8 i n i t p r i n t i n g ( )9

22

10 # Written with bad va r i ab l e names to match notat ion in Hewitt and Ross11 de f r ep r e s e l em ( j , k , l ,12 alpha=Symbol ( ’ alpha ’ , complex=True ) ,13 beta=Symbol ( ’ beta ’ , complex=True ) ) :14 # Have to do t h i s because numpy i sn ’ t p lay ing n i c e l y with negat ive exponents .15 pre f i x exp power = in t ( j−k )16 p r e f i x = (−1)∗∗( p re f i x exp power ) ∗ sq r t ( ( f a c t o r i a l ( l+j )∗ f a c t o r i a l ( l−j ) /( f a c t o r i a l ( l+k )∗ f a c t o r i a l ( l−k ) ) ) )17 # Setup vars f o r i t e r a t i o n /sum i t e r a t i o n18 upper=min ( l−j , l+k )19 lower=max(0 , k−j )20 # Run i t e r a t i o n ( summation )21 summand=022 f o r s in arange ( lower , upper+1 ,1) :23 summand += (−1)∗∗ s ∗ binomial ( l+k , s ) ∗ binomial ( l−k , l−j−s ) ∗ alpha ∗∗( l−j−s ) ∗ conj ( alpha ) ∗∗( l+k−s )

∗ beta ∗∗( s ) ∗ conj ( beta ) ∗∗( j−k+s )24 # Done25 return p r e f i x ∗summand2627 ##########################28 l =5/229 ##########################30 R = zero s ( i n t (2∗ l +1) )31 elems =[ ]32 f o r j in arange(− l , l +1 ,1) :33 row=[]34 f o r k in arange(− l , l +1 ,1) :35 row . append ( r ep re s e l em ( j , k , l ) )36 elems . append ( row )3738 R = Matrix ( elems )39 d i sp l ay (R)40 pr in t ( l a t ex (R) )

10.3 Base Code for Finite Group Fourier Transform

1 #! / usr /bin /python3 . 7 −−23 import numpy as np45 ’ ’ ’6 Custom matrix c l a s s that has the a t t r i b u t e s o f equa l i t y and not−equa l i t y that7 I want , which i s element−wise .8 ’ ’ ’9 c l a s s matrix (np . matrix ) :

10 de f e q ( s e l f , other ) :11 i f type ( other ) == type (None ) :12 return False13 A = s e l f14 B = other15 i f A. shape == B. shape :16 f o r i in range (A. shape [ 0 ] ) :17 f o r j in range (A. shape [ 1 ] ) :18 i f A[ i , j ] != B[ i , j ] :19 return False20 e l s e :21 r a i s e ’ Matrix Shape Error ’22 return True2324 de f n e ( s e l f , other ) :25 i f type ( other ) == type (None ) :26 return True27 A = s e l f28 B = other29 i f A. shape == B. shape :30 f o r i in range (A. shape [ 0 ] ) :31 f o r j in range (A. shape [ 1 ] ) :32 i f A[ i , j ] != B[ i , j ] :33 return True34 e l s e :35 r a i s e ’ Matrix Shape Error ’36 return False373839 ################################################################################40 c l a s s f i n i t e g r o up ( ) :4142 de f i n i t ( s e l f , elems , ac t i on ) :43 s e l f . elems = elems44 s e l f . a c t i on = act i on45 s e l f . c a r d i n a l i t y = len ( s e l f . e lems )46 i f s e l f . i s mat r i x g roup ( ) :47 s e l f . matr ix group = True48 e l s e :49 s e l f . matr ix group = False50 i f s e l f . i s g r oup ( ) i s Fa l se :51 s e l f . elems = None52 s e l f . a c t i on = None53 r a i s e ’ IsNotGroup ’5455 ’ ’ ’56 True : Al l e lements are matr i ces

23

57 False : At l e a s t one element i s not a matrix in the s e t o f e lements58 ’ ’ ’59 de f i s mat r i x g roup ( s e l f ) :60 f o r g in s e l f . elems :61 i f type ( g ) == type (np . matrix ( [ 0 ] ) ) :62 cont inue63 e l s e :64 return False65 return True6667 ’ ’ ’68 S a t i s f i e s a l l o f the group s t ru c tu r e p rope r t i e s , namely the ex i s t en c e o f an69 iden t i ty , e x i s t en c e o f i n v e r s e s and c l o s u r e .70 ’ ’ ’71 de f i s g r oup ( s e l f ) :72 i f s e l f . h a s i d en t i t y ( ) and s e l f . h a s c l o s u r e ( ) and s e l f . h a s i n v e r s e s ( ) :73 return True74 e l s e :75 return False7677 ’ ’ ’78 I f ac t i on (g , h) == act i on (h , g ) f o r a l l g and h , then the group i s abe l ian ,79 meaning that the group has a commutative ac t i on .80 ’ ’ ’81 de f i s commutat ive ( s e l f ) :82 X = s e l f . elems83 f o r g in X:84 f o r h in X:85 i f s e l f . a c t i on (g , h) != s e l f . a c t i on (h , g ) :86 return False87 e l s e :88 cont inue89 return True9091 de f h a s i d en t i t y ( s e l f ) :92 X = s e l f . elems93 f o r g in X:94 f o r h in X:95 # Found i d en t i t y candidate96 i f s e l f . a c t i on (g , h) == h :97 # Confirming i d en t i t y candidate98 f o r j in X:99 i f s e l f . a c t i on (g , j ) == j :

100 cont inue101 e l s e :102 break103 s e l f . i d e n t i t y = g104 return True105 return False106107 ’ ’ ’108 Desc r ip t i on :109 For a l l g , h in the group act i on (g , h) should be in the group as we l l .110 Otherwise , the s e t and act i on do not s a t i s f y the requirements o f a group111 and in s t ead are a groupoid .112 Return Values :113 True : i f i t does have c l o s u r e114 False : i f i t does not have c l o s u r e115 ’ ’ ’116 de f h a s c l o s u r e ( s e l f ) :117 X = s e l f . elems118 f o r g in X:119 f o r h in X:120 i f s e l f . a c t i on (g , h) not in X:121 return False122 return True123124 de f h a s i n v e r s e s ( s e l f ) :125 X = s e l f . elems126 f o r g in X:127 r e s u l t = s e l f . g e t i n v e r s e ( g )128 i f r e s u l t == None :129 return False130 return True131132 ’ ’ ’133 Get i nv e r s e o f a c e r t a i n element134 ’ ’ ’135 de f g e t i n v e r s e ( s e l f , e l ) :136 X = s e l f . elems137 f o r g in X:138 i f s e l f . a c t i on (g , e l ) == s e l f . i d e n t i t y :139 return g140 # Inve r s e not found141 return None142143 de f c ente r ( s e l f ) :144 pass145146 de f g e t i d e n t i t y ( s e l f ) :147 return s e l f . i d e n t i t y148149 ################################################################################

24

150151 de f main ( ) :152 x = matrix ( [153 [ 0 , 0 , 1 ] ,154 [ 1 , 0 , 0 ] ,155 [ 0 , 1 , 0 ] ] )156 y = matrix ( [157 [ 0 , 1 , 0 ] ,158 [ 1 , 0 , 0 ] ,159 [ 0 , 0 , 1 ] ] )160 I = matrix ( [161 [ 1 , 0 , 0 ] ,162 [ 0 , 1 , 0 ] ,163 [ 0 , 0 , 1 ] ] )164165 pr in t ( type (x ) )166 my set = [ I , x , y , x∗y , x∗x∗y , x∗x ]167 de f my action (a , b) :168 return a∗b169 G = f i n i t e g r o up (my set , my action )170 pr in t (G. i s g r oup ( ) )171 pr in t (G. g e t i d e n t i t y ( ) )172 pr in t (G. g e t i n v e r s e (x )∗x )173 pr in t ( I . t r a c e ( ) )174175 i f name == ’ ma in ’ :176 main ( )

11 Background

11.1 Arriving at the Fourier Transform

Fourier series are useful for periodic functions, but what about transforming those that areaperiodic? For example, if we look at the classic example of the rectangle function

it takes a value of +1 over −1/2 to 1/2 and then never repeats. The period of such a function mustbe the whole real line. We then have to extend our fundamental period to be infinitely long, i.e.1/ξ0 −→∞. But first we need to slightly rearrange the coefficient equation

Gn =1

ξ0

∫ 1/(2ξ0)

−1/(2ξ0)f(x)e−2πnξ0x dx −→ Gnξ0 =

∫ 1/(2ξ0)

−1/(2ξ0)f(x)e−2πnξ0x dx

and we now can extend the period by taking the limit of both sides

lim1/ξ0→∞

(Gnξ0

)= lim

1/ξ0→∞

(∫ 1/(2ξ0)

−1/(2ξ0)f(x)e−2πnξ0x dx

)nξ0 −→ ξ

1

2ξ0−→∞ −1

2ξ0−→ −∞ Gnξ0 −→ f(ξ)

25

then substituting our new values, we have arrived at the definition of the Fourier transform

f(ξ) =

∫ ∞−∞

f(x)e−2πiξx dx.

11.2 The Fourier Transform and its Properties

We first turn our attention to the original definition of the transformation of functions over the realline.

F(f) = f(ξ) =

∫ ∞−∞

f(x)e−2πiξx dx

There are a few components to this transformation that we can examine, first the function f is onethat maps from the real-line to the complex numbers

f : R −→ C.

e−2πiξx is the kernel8 of the transform and dx is the Lebesgue measure [Tao, 2016].

The results below can be derived using single variable calculus. The consequences in engineering ofthese results can be discussed at great length, though we leave that to other authors such asOsgood [Osgood, 2019].

11.3 General Properties

11.3.1 Convolution

Convolution is an operation on two functions f and g where we calculate∫ ∞−∞

f(x− τ)g(τ) dτ.

Convolution in one domain leads to multiplication in the codomain, explicitly this means

F(f ∗ g) = F(∫ ∞−∞

f(x− τ)g(τ) dτ

)= f g

and

F−1(f ∗ g) = F−1

(∫ ∞−∞

f(ξ − τ)g(τ) dτ

)= f g.

11.3.2 Translation Effects

A delay in the domain (here, the real line R) leads to a multiplication of a function in the codomain.

F(f(x− a)) =

∫Rf(x− a)e−2πiξx dx = e2πiξ(−a)f(ξ)

8Note that when we use kernel what is meant is the kernel of the integral transformation and not an algebraichomomorphism mapping into the identity.

26

11.3.3 Parseval-Plancherel Theorem

For periodic functions with a period P where

P :=

[−1

2ξ0,

1

2ξ0

]⊆ R

∥∥∥f∥∥∥2

L2(P )=

∫ ∞−∞|f(x)|2 dx =

∞∑n=−∞

|Gn|2

For aperiodic functions: ∥∥∥f∥∥∥2

L2(R)=

∫ ∞−∞|f(x)|2 dx =

∫ ∞−∞|f(ξ)|2 dξ

Proof and exposition of these theorems can be found in Folland [Folland, 2015].

11.3.4 Orthogonality Relations

For a Fourier series of a function, we get something that looks like

f(x) = . . .+G0 +G1eix +G2e

i2x +G3ei3x + . . .

we often refer to the einx term as a harmonic. In later parts of this report the orthogonality ofcoefficients of a Fourier series will be important to link our idea of harmonic analysis to Fourieranalysis and so we shall demonstrate the orthogonality of harmonics here.

We always test for orthogonality with the inner product:∫f g dx.

If the integral is zero, then f and g are orthogonal. Now let us suppose we have two differentharmonics, einx and eimx, where n−m 6= 0 and we are testing for orthogonality over [0, 2π], then∫ 2π

0einxeimx dx =

∫ 2π

0ei(n−m)x dx =

ei(n−m)2π − 1

i(n−m)= 0 ∀ n,m ∈ Z | n−m 6= 0

and if n−m = 0? Then n = m and so we have∫ 2π

0ei(m−m)x dx =

∫ 2π

0dx = 2π ∀ n,m ∈ Z | n−m = 0

in summary, for integers n and m:∫ 2π

0einxeimx dx =

{2π n = m

0 n 6= m

27

11.4 Some Linear Algebra

11.4.1 Basis

A basis of a vector space V is a set of vectors that are linearly independent and span the space. Oneof the simplest examples is e1, e2 and e3 that we use in a 3-dimensional vector space. This basis (andevery basis) is simply a set of vectors we can use to express all other vectors in the form of linearcombinations. i.e.

∀ v ∈ V v = a1e1 + . . .+ anen | a1, . . . , an ∈ R.

Notation can vary, however. For example, if we denote our basis B and denote each column byb1, . . . ,bn then we get:

B =

b11 . . . b1n...

. . ....

bn1 . . . bnn

=[b1 . . .bn

]where our example of a three dimensional vector space with the standard basis would take the form

B3 =[e1 e2 e3

]=

100

010

001

=

1 0 00 1 00 0 1

[Strang, 2006].

11.4.2 Orthonormal Basis

An orthonormal basis is one that is both orthogonal and normalised. Orthogonality is where eachbi ⊥ bj for all i 6= j, i.e. bi · bj = 0 for all vectors not the same. Normalisation means that thelength of each basis vector is one, |b| = 1 [Strang, 2006].

11.4.3 Trace Operator

Given a matrix A of size n× n, we can take the diagonal entries and sum them, this is called takingthe trace of A. More explicitly, if we denote Aij as the entry of A on its i-th row and j-th column,then the trace is

trace(A) =

n∑i=1

Aii.

Trace is invariant of basis [Strang, 2006] and should it be performed on a matrix group then theoperation is constant over the group’s conjugacy classes [Artin, 2010].

11.4.4 Groups of Matrices

GL(n) is the Lie group of all n× n invertible matrices9.

∀A ∈ GL(n) ∃ A−1 ∈ G | A−1A = AA−1 = I

9Recall that not all matrices have inverses and so GL(n) is a subset of all the matrices that exist.

28

U(n) denotes the Lie group of unitary matrices of size n× n, who have the property that U∗ = U−1

where the ∗ operation indicates Hermitian conjugate,

UU∗ = U∗U = I.

O(n) is the Lie group of orthogonal matrices of size n× n who have the property A−1 = AT ,

AAT = ATA = I.

11.5 Duality

11.5.1 Duality in Sets

The complement of a set A is all the elements not in A, and is usually denoted by A.

Observe that the operation is an involution, i.e. it is its own inverse (like the function 1/x or −x).Because of this, the complement of A is A but the complement of A is A, more explicitly

A = A.

This inherent relationship between the two is one of the simplest and illustrative examples of theprinciple of duality.

11.5.2 Duality in Vector Spaces

Consider the vector space V of length n column vectors. What other space could act on this spaceand send it to the scalars? Only the space of all length n row vectors, which we will call V ∗. Thenfor v ∈ V and v∗ ∈ V ∗

v∗v =[v∗1 . . . v∗n

] v1...vn

= v1v∗1 + . . .+ vnv

∗n =

n∑i

viv∗i .

This gives us a way to produce a scalar from any v and v∗ from their respective spaces.

We say that V ∗ is the space of linear mappings to R on V , and that V is the space of linearmappings to R on V ∗. V and V ∗ are termed dual vector spaces of each other.

11.6 Miscellaneous

Below are some other frequently used pieces of mathematics that don’t easily fit into any othersection.

11.6.1 Direct Sum (⊕)

Say we have two matrices A and B defined by

A =

[a11 a12

a21 a22

]B =

[b11 b12

b21 b22

]29

then their direct sum A⊕B is

A⊕B =

[A 00 B

]=

a11 a12 0 0a12 a22 0 00 0 b11 b12

0 0 b21 b22

.At times, direct sums involve many matrices and take a form similar to the Σ notation often used forarithmetic summation, for example

n⊕i=1

Ai =

A1 0 . . . 00 A2 . . . 0

0...

. . ....

0 0 . . . An

.Group representations are often made up of smaller representations composed with a direct sum.

11.6.2 Tensor Product (⊗)

For a vector v ∈ Rn and w ∈ Rm the tensor product between them is

v ⊗ w =

v1

v2...vn

⊗w1

w2...wn

=

v1w1 v1w2 . . . v1wmv2w1 v2w2 . . . v2wm

......

. . ....

vnw1 vnw2 . . . vnwm

[Neuenschwander, 2015].

11.6.3 Permutation Notation

A permutation is a bijective function from a set onto itself. Consider a sequence of values{x1, x2, x3, x4}, we can rearrange these in the order {x4, x1, x2, x3} but how do we state thismathematically? Perhaps as a mapping

{x1, x2, x3, x} 7−→ {x4, x1, x2, x3}

although the convention is often to use permutation matrices or permutation notation. Apermutation matrix for our rearrangement would look like

0 0 0 11 0 0 00 1 0 00 0 1 0

and permutation notation would take the form (1 2 3 4), [Artin, 2010]. The meaning of thisbracketed notation is that element 1 goes to place 2, element 2 goes to place 3, element 3 goes toplace 4 and element 4 goes to place 1. It is easy to see why this notation is used when we examinehow brief it is to state (3 32) instead of writing a 32× 32 permutation matrix (assuming 32 elementsin the set).

30

11.6.4 Dense Sets

For a topological space X, a subset A ⊂ X is called dense if every point in X belongs to A or is alimit point of A [Munkres, 2000].

31