# Holomorphic Logarithms and Roots

In this post we’ll make sense of a holomorphic square root and logarithm. Wrote this up because I was surprised how hard it was to find a decent complete explanation.

Let ${f : U \rightarrow \mathbb C}$ be a holomorphic function. A holomorphic ${n}$th root of ${f}$ is a function ${g : U \rightarrow \mathbb C}$ such that ${f(z) = g(z)^n}$ for all ${z \in U}$. A logarithm of ${f}$ is a function ${g : U \rightarrow \mathbb C}$ such that ${f(z) = e^{g(z)}}$ for all ${z \in U}$. The main question we’ll try to figure out is: when do these exist? In particular, what if ${f = \mathrm{id}}$?

## 1. Motivation: Square Root of a Complex Number

To start us off, can we define ${\sqrt z}$ for any complex number ${z}$?

The first obvious problem that comes up is that there for any ${z}$, there are two numbers ${w}$ such that ${w^2 = z}$. How can we pick one to use? For our ordinary square root function, we had a notion of “positive”, and so we simply took the positive root.

Let’s expand on this: given ${ z = r \left( \cos\theta + i \sin\theta \right) }$ (here ${r \ge 0}$) we should take the root to be

$\displaystyle w = \sqrt{r} \left( \cos \alpha + i \sin \alpha \right).$

such that ${2\alpha \equiv \theta \pmod{2\pi}}$; there are two choices for ${\alpha \pmod{2\pi}}$, differing by ${\pi}$.

For complex numbers, we don’t have an obvious way to pick ${\alpha}$. Nonetheless, perhaps we can also get away with an arbitrary distinction: let’s see what happens if we just choose the ${\alpha}$ with ${-\frac{1}{2}\pi < \alpha \le \frac{1}{2}\pi}$.

Pictured below are some points (in red) and their images (in blue) under this “upper-half” square root. The condition on ${\alpha}$ means we are forcing the blue points to lie on the right-half plane.

Here, ${w_i^2 = z_i}$ for each ${i}$, and we are constraining the ${w_i}$ to lie in the right half of the complex plane. We see there is an obvious issue: there is a big discontinuity near the point ${z_5}$ and ${z_7}$! The nearby point ${w_6}$ has been mapped very far away. This discontinuity occurs since the points on the negative real axis are at the “boundary”. For example, given ${-4}$, we send it to ${-2i}$, but we have hit the boundary: in our interval ${-\frac{1}{2}\pi \le \alpha < \frac{1}{2}\pi}$, we are at the very left edge.

The negative real axis that we must not touch is is what we will later call a branch cut, but for now I call it a ray of death. It is a warning to the red points: if you cross this line, you will die! However, if we move the red circle just a little upwards (so that it misses the negative real axis) this issue is avoided entirely, and we get what seems to be a “nice” square root.

In fact, the ray of death is fairly arbitrary: it is the set of “boundary issues” that arose when we picked ${-\frac{1}{2}\pi < \alpha \le \frac{1}{2}\pi}$. Suppose we instead insisted on the interval ${0 \le \alpha < \pi}$; then the ray of death would be the positive real axis instead. The earlier circle we had now works just fine.

What we see is that picking a particular ${\alpha}$-interval leads to a different set of edge cases, and hence a different ray of death. The only thing these rays have in common is their starting point of zero. In other words, given a red circle and a restriction of ${\alpha}$, I can make a nice “square rooted” blue circle as long as the ray of death misses it.

So, what exactly is going on?

## 2. Square Roots of Holomorphic Functions

To get a picture of what’s happening, we would like to consider a more general problem: let ${f: U \rightarrow \mathbb C}$ be holomorphic. Then we want to decide whether there is a ${g : U \rightarrow \mathbb C}$ such that

$\displaystyle f(z) = g(z)^2.$

Our previous discussion when ${f = \mathrm{id}}$ tells us we cannot hope to achieve this for ${U = \mathbb C}$; there is a “half-ray” which causes problems. However, there are certainly functions ${f : \mathbb C \rightarrow \mathbb C}$ such that a ${g}$ exists. As a simplest example, ${f(z) = z^2}$ should definitely have a square root!

Now let’s see if we can fudge together a square root. Earlier, what we did was try to specify a rule to force one of the two choices at each point. This is unnecessarily strict. Perhaps we can do something like the following: start at a point in ${z_0 \in U}$, pick a square root ${w_0}$ of ${f(z_0)}$, and then try to “fudge” from there the square roots of the other points. What do I mean by fudge? Well, suppose ${z_1}$ is a point very close to ${z_0}$, and we want to pick a square root ${w_1}$ of ${f(z_1)}$. While there are two choices, we also would expect ${w_0}$ to be close to ${w_1}$. Unless we are highly unlucky, this should tells us which choice of ${w_1}$ to pick. (Stupid concrete example: if I have taken the square root ${-4.12i}$ of ${-17}$ and then ask you to continue this square root to ${-16}$, which sign should you pick for ${\pm 4i}$?)

There are two possible ways we could get unlucky in the scheme above: first, if ${w_0 = 0}$, then we’re sunk. But even if we avoid that, we have to worry that we are in a situation, where we run around a full loop in the complex plane, and then find that our continuous perturbation has left us in a different place than we started. For concreteness, consider the following situation, again with ${f = \mathrm{id}}$:

We started at the point ${z_0}$, with one of its square roots as ${w_0}$. We then wound a full red circle around the origin, only to find that at the end of it, the blue arc is at a different place where it started!

The interval construction from earlier doesn’t work either: no matter how we pick the interval for ${\alpha}$, any ray of death must hit our red circle. The problem somehow lies with the fact that we have enclosed the very special point ${0}$.

Nevertheless, we know that if we take ${f(z) = z^2}$, then we don’t run into any problems with our “make it up as you go” procedure. So, what exactly is going on?

## 3. Covering Projections

By now, if you have read the part of algebraic topology. this should all seem very strangely familiar. The “fudging” procedure exactly describes the idea of a lifting.

More precisely, recall that there is a covering projection

$\displaystyle (-)^2 : \mathbb C \setminus \{0\} \rightarrow \mathbb C \setminus \{0\}.$

Let ${V = \left\{ z \in U \mid f(z) \neq 0 \right\}}$. For ${z \in U \setminus V}$, we already have the square root ${g(z) = \sqrt{f(z)} = \sqrt 0 = 0}$. So the burden is completing ${g : V \rightarrow \mathbb C}$.

Then essentially, what we are trying to do is construct a lifting ${g}$ for the following diagram: Our map ${p}$ can be described as “winding around twice”. From algebraic topology, we now know that this lifting exists if and only if

$\displaystyle f_\ast(\pi_1(V)) \subseteq p_\ast(\pi_1(E))$

is a subset of the image of ${\pi_1(E)}$ by ${p}$. Since ${B}$ and ${E}$ are both punctured planes, we can identify them with ${S^1}$.

Ques 1

Show that the image under ${p}$ is exactly ${2\mathbb Z}$ once we identify ${\pi_1(B) = \mathbb Z}$.

That means that for any loop ${\gamma}$ in ${V}$, we need ${f \circ \gamma}$ to have an even winding number around ${0 \in B}$. This amounts to

$\displaystyle \frac{1}{2\pi} \oint_\gamma \frac{f'}{f} \; dz \in 2\mathbb Z$

since ${f}$ has no poles.

Replacing ${2}$ with ${n}$ and carrying over the discussion gives the first main result.

Theorem 2 (Existence of Holomorphic ${n}$th Roots)

Let ${f : U \rightarrow \mathbb C}$ be holomorphic. Then ${f}$ has a holomorphic ${n}$th root if and only if

$\displaystyle \frac{1}{2\pi i}\oint_\gamma \frac{f'}{f} \; dz \in n\mathbb Z$

for every contour ${\gamma}$ in ${U}$.

## 4. Complex Logarithms

The multivalued nature of the complex logarithm comes from the fact that

$\displaystyle \exp(z+2\pi i) = \exp(z).$

So if ${e^w = z}$, then any complex number ${w + 2\pi i k}$ is also a solution.

We can handle this in the same way as before: it amounts to a lifting of the following diagram. There is no longer a need to work with a separate ${V}$ since:

Ques 3

Show that if ${f}$ has any zeros then ${g}$ possibly can’t exist.

In fact, the map ${\exp : \mathbb C \rightarrow \mathbb C\setminus\{0\}}$ is a universal cover, since ${\mathbb C}$ is simply connected. Thus, ${p(\pi_1(\mathbb C))}$ is trivial. So in addition to being zero-free, ${f}$ cannot have any winding number around ${0 \in B}$ at all. In other words:

Theorem 4 (Existence of Logarithms)

Let ${f : U \rightarrow \mathbb C}$ be holomorphic. Then ${f}$ has a logarithm if and only if

$\displaystyle \frac{1}{2\pi i}\oint_\gamma \frac{f'}{f} \; dz = 0$

for every contour ${\gamma}$ in ${U}$.

## 5. Some Special Cases

The most common special case is

Corollary 5 (Nonvanishing Functions from Simply Connected Domains)

Let ${f : \Omega \rightarrow \mathbb C}$ be continuous, where ${\Omega}$ is simply connected. If ${f(z) \neq 0}$ for every ${z \in \Omega}$, then ${f}$ has both a logarithm and holomorphic ${n}$th root.

Finally, let’s return to the question of ${f = \mathrm{id}}$ from the very beginning. What’s the best domain ${U}$ such that we can define ${\sqrt{-} : U \rightarrow \mathbb C}$? Clearly ${U = \mathbb C}$ cannot be made to work, but we can do almost as well. For note that the only zero of ${f = \mathrm{id}}$ is at the origin. Thus if we want to make a logarithm exist, all we have to do is make an incision in the complex plane that renders it impossible to make a loop around the origin. The usual choice is to delete negative half of the real axis, our very first ray of death; we call this a branch cut, with branch point at ${0 \in \mathbb C}$ (the point which we cannot circle around). This gives

Theorem 6 (Branch Cut Functions)

There exist holomorphic functions

\displaystyle \begin{aligned} \log &: \mathbb C \setminus (-\infty, 0] \rightarrow \mathbb C \\ \sqrt[n]{-} &: \mathbb C \setminus (-\infty, 0] \rightarrow \mathbb C \end{aligned}

satisfying the obvious properties.

There are many possible choices of such functions (${n}$ choices for the ${n}$th root and infinitely many for ${\log}$); a choice of such a function is called a branch. So this is what is meant by a “branch” of a logarithm.

The principal branch is the “canonical” branch, analogous to the way we arbitrarily pick the positive branch to define ${\sqrt{-} : \mathbb R_{\ge 0} \rightarrow \mathbb R_{\ge 0}}$. For ${\log}$, we take the ${w}$ such that ${e^w = z}$ and the imaginary part of ${w}$ lies in ${(-\pi, \pi]}$ (since we can shift by integer multiples of ${2\pi i}$). Often, authors will write ${\text{Log } z}$ to emphasize this choice.

Example 7

Let ${U}$ be the complex plane minus the real interval ${[0,1]}$. Then the function ${U \rightarrow \mathbb C}$ by ${z \mapsto z(z-1)}$ has a holomorphic square root.

Corollary 8

A holomorphic function ${f : U \rightarrow \mathbb C}$ has a holomorphic ${n}$th root for all ${n \ge 1}$ if and only if it has a holomorphic logarithm.

# Things Fourier

For some reason several classes at MIT this year involve Fourier analysis. I was always confused about this as a high schooler, because no one ever gave me the “orthonormal basis” explanation, so here goes. As a bonus, I also prove a form of Arrow’s Impossibility Theorem using binary Fourier analysis, and then talk about the fancier generalizations using Pontryagin duality and the Peter-Weyl theorem.

In what follows, we let ${\mathbb T = \mathbb R/\mathbb Z}$ denote the “circle group”, thought of as the additive group of “real numbers modulo ${1}$”. There is a canonical map ${e : \mathbb T \rightarrow \mathbb C}$ sending ${\mathbb T}$ to the complex unit circle, given by ${e(\theta) = \exp(2\pi i \theta)}$.

Disclaimer: I will deliberately be sloppy with convergence issues, in part because I don’t fully understand them myself, and in part because I don’t care.

## 1. Synopsis

Suppose we have a domain ${Z}$ and are interested in functions ${f : Z \rightarrow \mathbb C}$. Naturally, the set of such functions form a complex vector space. We like to equip the set of such functions with an positive definite inner product. The idea of Fourier analysis is to then select an orthonormal basis for this set of functions, say ${(e_\xi)_{\xi}}$, which we call the characters; the indexing ${\xi}$ are called frequencies. In that case, since we have a basis, every function ${f : Z \rightarrow \mathbb C}$ becomes a sum

$\displaystyle f(x) = \sum_{\xi} \widehat f(\xi) e_\xi$

where ${\widehat f(\xi)}$ are complex coefficients of the basis; appropriately we call ${\widehat f}$ the Fourier coefficients. The variable ${x \in Z}$ is referred to as the physical variable. This is generally good because the characters are deliberately chosen to be nice “symmetric” functions, like sine or cosine waves or other periodic functions. Thus ${we}$ decompose an arbitrarily complicated function into a sum on nice ones.

For convenience, we record a few facts about orthonormal bases.

Proposition 1 (Facts about orthonormal bases)

Let ${V}$ be a complex Hilbert space with inner form ${\left< -,-\right>}$ and suppose ${x = \sum_\xi a_\xi e_\xi}$ and ${y = \sum_\xi b_\xi e_\xi}$ where ${e_\xi}$ are an orthonormal basis. Then

\displaystyle \begin{aligned} \left< x,x \right> &= \sum_\xi |a_\xi|^2 \\ a_\xi &= \left< x, e_\xi \right> \\ \left< x,y \right> &= \sum_\xi a_\xi \overline{b_\xi}. \end{aligned}

## 2. Common Examples

### 2.1. Binary Fourier analysis on ${\{\pm1\}^n}$

Let ${Z = \{\pm 1\}^n}$ for some positive integer ${n}$, so we are considering functions ${f(x_1, \dots, x_n)}$ accepting binary values. Then the functions ${Z \rightarrow \mathbb C}$ form a ${2^n}$-dimensional vector space ${\mathbb C^Z}$, and we endow it with the inner form

$\displaystyle \left< f,g \right> = \frac{1}{2^n} \sum_{x \in Z} f(x) \overline{g(x)}.$

In particular,

$\displaystyle \left< f,f \right> = \frac{1}{2^n} \sum_{x \in Z} \left\lvert f(x) \right\rvert^2$

is the average of the squares; this establishes also that ${\left< -,-\right>}$ is positive definite.

In that case, the multilinear polynomials form a basis of ${\mathbb C^Z}$, that is the polynomials

$\displaystyle \chi_S(x_1, \dots, x_n) = \prod_{s \in S} x_s.$

Thus our frequency set is actually the subsets ${S \subseteq \{1, \dots, n\}}$. Thus, we have a decomposition

$\displaystyle f = \sum_{S \subseteq \{1, \dots, n\}} \widehat f(S) \chi_S.$

Example 2 (An example of binary Fourier analysis)

Let ${n = 2}$. Then binary functions ${\{ \pm 1\}^2 \rightarrow \mathbb C}$ have a basis given by the four polynomials

$\displaystyle 1, \quad x_1, \quad x_2, \quad x_1x_2.$

For example, consider the function ${f}$ which is ${1}$ at ${(1,1)}$ and ${0}$ elsewhere. Then we can put

$\displaystyle f(x_1, x_2) = \frac{x_1+1}{2} \cdot \frac{x_2+1}{2} = \frac14 \left( 1 + x_1 + x_2 + x_1x_2 \right).$

So the Fourier coefficients are ${\widehat f(S) = \frac 14}$ for each of the four ${S}$‘s.

This notion is useful in particular for binary functions ${f : \{\pm1\}^n \rightarrow \{\pm1\}}$; for these functions (and products thereof), we always have ${\left< f,f \right> = 1}$.

It is worth noting that the frequency ${\varnothing}$ plays a special role:

Exercise 3

Show that

$\displaystyle \widehat f(\varnothing) = \frac{1}{|Z|} \sum_{x \in Z} f(x).$

### 2.2. Fourier analysis on finite groups ${Z}$

This is the Fourier analysis used in this post and this post. Here, we have a finite abelian group ${Z}$, and consider functions ${Z \rightarrow \mathbb C}$; this is a ${|Z|}$-dimensional vector space. The inner product is the same as before:

$\displaystyle \left< f,g \right> = \frac{1}{|Z|} \sum_{x \in Z} f(x) \overline{g}(x).$

Now here is how we generate the characters. We equip ${Z}$ with a non-degenerate symmetric bilinear form

$\displaystyle Z \times Z \xrightarrow{\cdot} \mathbb T \qquad (\xi, x) \mapsto \xi \cdot x.$

Experts may already recognize this as a choice of isomorphism between ${Z}$ and its Pontryagin dual. This time the characters are given by

$\displaystyle \left( e_\xi \right)_{\xi \in Z} \qquad \text{where} \qquad e_\xi(x) = e(\xi \cdot x).$

In this way, the set of frequencies is also ${Z}$, but the ${\xi \in Z}$ play very different roles from the “physical” ${x \in Z}$. (It is not too hard to check these indeed form an orthonormal basis in the function space ${\mathbb C^{\left\lvert Z \right\rvert}}$, since we assumed that ${\cdot}$ is non-degenerate.)

Example 4 (Cube roots of unity filter)

Suppose ${Z = \mathbb Z/3\mathbb Z}$, with the inner form given by ${\xi \cdot x = (\xi x)/3}$. Let ${\omega = \exp(\frac 23 \pi i)}$ be a primitive cube root of unity. Note that

$\displaystyle e_\xi(x) = \begin{cases} 1 & \xi = 0 \\ \omega^x & \xi = 1 \\ \omega^{2x} & \xi = 2. \end{cases}$

Then given ${f : Z \rightarrow \mathbb C}$ with ${f(0) = a}$, ${f(1) = b}$, ${f(2) = c}$, we obtain

$\displaystyle f(x) = \frac{a+b+c}{3} \cdot 1 + \frac{a + \omega^2 b + \omega c}{3} \cdot \omega^x + \frac{a + \omega b + \omega^2 c}{3} \cdot \omega^{2x}.$

In this way we derive that the transforms are

\displaystyle \begin{aligned} \widehat f(0) &= \frac{a+b+c}{3} \\ \widehat f(1) &= \frac{a+\omega^2 b+ \omega c}{3} \\ \widehat f(2) &= \frac{a+\omega b+\omega^2c}{3}. \end{aligned}

Exercise 5

Show that

$\displaystyle \widehat f(0) = \frac{1}{|Z|} \sum_{x \in Z} f(x).$

Olympiad contestants may recognize the previous example as a “roots of unity filter”, which is exactly the point. For concreteness, suppose one wants to compute

$\displaystyle \binom{1000}{0} + \binom{1000}{3} + \dots + \binom{1000}{999}.$

In that case, we can consider the function

$\displaystyle w : \mathbb Z/3 \rightarrow \mathbb C.$

such that ${w(0) = 1}$ but ${w(1) = w(2) = 0}$. By abuse of notation we will also think of ${w}$ as a function ${w : \mathbb Z \twoheadrightarrow \mathbb Z/3 \rightarrow \mathbb C}$. Then the sum in question is

\displaystyle \begin{aligned} \sum_n \binom{1000}{n} w(n) &= \sum_n \binom{1000}{n} \sum_{k=0,1,2} \widehat w(k) \omega^{kn} \\ &= \sum_{k=0,1,2} \widehat w(k) \sum_n \binom{1000}{n} \omega^{kn} \\ &= \sum_{k=0,1,2} \widehat w(k) (1+\omega^k)^n. \end{aligned}

In our situation, we have ${\widehat w(0) = \widehat w(1) = \widehat w(2) = \frac13}$, and we have evaluated the desired sum. More generally, we can take any periodic weight ${w}$ and use Fourier analysis in order to interchange the order of summation.

Example 6 (Binary Fourier analysis)

Suppose ${Z = \{\pm 1\}^n}$, viewed as an abelian group under pointwise multiplication hence isomorphic to ${(\mathbb Z/2\mathbb Z)^{\oplus n}}$. Assume we pick the dot product defined by

$\displaystyle \xi \cdot x = \frac{1}{2} \sum_i \xi_i x_i$

where ${\xi = (\xi_1, \dots, \xi_n)}$ and ${x = (x_1, \dots, x_n)}$.

We claim this coincides with the first example we gave. Indeed, let ${S \subseteq \{1, \dots, n\}}$ and let ${\xi \in \{\pm1\}^n}$ which is ${-1}$ at positions in ${S}$, and ${+1}$ at positions not in ${S}$. Then the character ${\chi_S}$ form the previous example coincides with the character ${e_\xi}$ in the new notation. In particular, ${\widehat f(S) = \widehat f(\xi)}$.

Thus Fourier analysis on a finite group ${Z}$ subsumes binary Fourier analysis.

### 2.3. Fourier series for functions ${L^2([-\pi, \pi])}$

Now we consider the space ${L^2([-\pi, \pi])}$ of square-integrable functions ${[-\pi, \pi] \rightarrow \mathbb C}$, with inner form

$\displaystyle \left< f,g \right> = \frac{1}{2\pi} \int_{[-\pi, \pi]} f(x) \overline{g(x)}.$

Sadly, this is not a finite-dimensional vector space, but fortunately it is a Hilbert space so we are still fine. In this case, an orthonormal basis must allow infinite linear combinations, as long as the sum of squares is finite.

Now, it turns out in this case that

$\displaystyle (e_n)_{n \in \mathbb Z} \qquad\text{where}\qquad e_n(x) = \exp(inx)$

is an orthonormal basis for ${L^2([-\pi, \pi])}$. Thus this time the frequency set ${\mathbb Z}$ is infinite. So every function ${f \in L^2([-\pi, \pi])}$ decomposes as

$\displaystyle f(x) = \sum_n \widehat f(n) \exp(inx)$

for ${\widehat f(n)}$.

This is a little worse than our finite examples: instead of a finite sum on the right-hand side, we actually have an infinite sum. This is because our set of frequencies is now ${\mathbb Z}$, which isn’t finite. In this case the ${\widehat f}$ need not be finitely supported, but do satisfy ${\sum_n |\widehat f(n)|^2 < \infty}$.

Since the frequency set is indexed by ${\mathbb Z}$, we call this a Fourier series to reflect the fact that the index is ${n \in \mathbb Z}$.

Exercise 7

Show once again

$\displaystyle \widehat f(0) = \frac{1}{2\pi} \int_{[-\pi, \pi]} f(x).$

Often we require that the function ${f}$ satisfies ${f(-\pi) = f(\pi)}$, so that ${f}$ becomes a periodic function, and we can think of it as ${f : \mathbb T \rightarrow \mathbb C}$.

### 2.4. Summary

We summarize our various flavors of Fourier analysis in the following table.

$\displaystyle \begin{array}{llll} \hline \text{Type} & \text{Physical var} & \text{Frequency var} & \text{Basis functions} \\ \hline \textbf{Binary} & \{\pm1\}^n & \text{Subsets } S \subseteq \left\{ 1, \dots, n \right\} & \prod_{s \in S} x_s \\ \textbf{Finite group} & Z & \xi \in Z, \text{ choice of } \cdot, & e(\xi \cdot x) \\ \textbf{Fourier series} & \mathbb T \text{ or } [-\pi, \pi] & n \in \mathbb Z & \exp(inx) \\ \end{array}$

In fact, we will soon see that all these examples are subsumed by Pontryagin duality for compact groups ${G}$.

## 3. Parseval and friends

The notion of an orthonormal basis makes several “big-name” results in Fourier analysis quite lucid. Basically, we can take every result from Proposition~1, translate it into the context of our Fourier analysis, and get a big-name result.

Corollary 8 (Parseval theorem)

Let ${f : Z \rightarrow \mathbb C}$, where ${Z}$ is a finite abelian group. Then

$\displaystyle \sum_\xi |\widehat f(\xi)|^2 = \frac{1}{|Z|} \sum_{x \in Z} |f(x)|^2.$

Similarly, if ${f : [-\pi, \pi] \rightarrow \mathbb C}$ is square-integrable then its Fourier series satisfies

$\displaystyle \sum_n |\widehat f(n)|^2 = \frac{1}{2\pi} \int_{[-\pi, \pi]} |f(x)|^2.$

Proof: Recall that ${\left< f,f\right>}$ is equal to the square sum of the coefficients. $\Box$

Corollary 9 (Formulas for ${\widehat f}$)

Let ${f : Z \rightarrow \mathbb C}$, where ${Z}$ is a finite abelian group. Then

$\displaystyle \widehat f(\xi) = \frac{1}{|Z|} \sum_{x \in Z} f(x) \overline{e_\xi(x)}.$

Similarly, if ${f : [-\pi, \pi] \rightarrow \mathbb C}$ is square-integrable then its Fourier series is given by

$\displaystyle \widehat f(n) = \frac{1}{2\pi} \int_{[-\pi, \pi]} f(x) \exp(-inx).$

Proof: Recall that in an orthonormal basis ${(e_\xi)_\xi}$, the coefficient of ${e_\xi}$ in ${f}$ is ${\left< f, e_\xi\right>}$. $\Box$
Note in particular what happens if we select ${\xi = 0}$ in the above!

Corollary 10 (Plancherel theorem)

Let ${f : Z \rightarrow \mathbb C}$, where ${Z}$ is a finite abelian group. Then

$\displaystyle \left< f,g \right> = \sum_{\xi \in Z} \widehat f(\xi) \overline{\widehat g(\xi)}.$

Similarly, if ${f : [-\pi, \pi] \rightarrow \mathbb C}$ is square-integrable then

$\displaystyle \left< f,g \right> = \sum_n \widehat f(\xi) \overline{\widehat g(\xi)}.$

Proof: Guess! $\Box$

## 4. (Optional) Arrow’s Impossibility Theorem

As an application, we now prove a form of Arrow’s theorem. Consider ${n}$ voters voting among ${3}$ candidates ${A}$, ${B}$, ${C}$. Each voter specifies a tuple ${v_i = (x_i, y_i, z_i) \in \{\pm1\}^3}$ as follows:

• ${x_i = 1}$ if ${A}$ ranks ${A}$ ahead of ${B}$, and ${x_i = -1}$ otherwise.
• ${y_i = 1}$ if ${A}$ ranks ${B}$ ahead of ${C}$, and ${y_i = -1}$ otherwise.
• ${z_i = 1}$ if ${A}$ ranks ${C}$ ahead of ${A}$, and ${z_i = -1}$ otherwise.

Tacitly, we only consider ${3! = 6}$ possibilities for ${v_i}$: we forbid “paradoxical” votes of the form ${x_i = y_i = z_i}$ by assuming that people’s votes are consistent (meaning the preferences are transitive).

Then, we can consider a voting mechanism

\displaystyle \begin{aligned} f : \{\pm1\}^n &\rightarrow \{\pm1\} \\ g : \{\pm1\}^n &\rightarrow \{\pm1\} \\ h : \{\pm1\}^n &\rightarrow \{\pm1\} \end{aligned}

such that ${f(x_\bullet)}$ is the global preference of ${A}$ vs. ${B}$, ${g(y_\bullet)}$ is the global preference of ${B}$ vs. ${C}$, and ${h(z_\bullet)}$ is the global preference of ${C}$ vs. ${A}$. We’d like to avoid situations where the global preference ${(f(x_\bullet), g(y_\bullet), h(z_\bullet))}$ is itself paradoxical.

In fact, we will prove the following theorem:

Theorem 11 (Arrow Impossibility Theorem)

Assume that ${(f,g,h)}$ always avoids paradoxical outcomes, and assume ${\mathbf E f = \mathbf E g = \mathbf E h = 0}$. Then ${(f,g,h)}$ is either a dictatorship or anti-dictatorship: there exists a “dictator” ${k}$ such that

$\displaystyle f(x_\bullet) = \pm x_k, \qquad g(y_\bullet) = \pm y_k, \qquad h(z_\bullet) = \pm z_k$

where all three signs coincide.

The “irrelevance of independent alternatives” reflects that The assumption ${\mathbf E f = \mathbf E g = \mathbf E h = 0}$ provides symmetry (and e.g. excludes the possibility that ${f}$, ${g}$, ${h}$ are constant functions which ignore voter input). Unlike the usual Arrow theorem, we do not assume that ${f(+1, \dots, +1) = +1}$ (hence possibility of anti-dictatorship).

To this end, we actually prove the following result:

Lemma 12

Assume the ${n}$ voters vote independently at random among the ${3! = 6}$ possibilities. The probability of a paradoxical outcome is exactly

$\displaystyle \frac14 + \frac14 \sum_{S \subseteq \{1, \dots, n\}} \left( -\frac13 \right)^{\left\lvert S \right\rvert} \left( \widehat f(S) \widehat g(S) + \widehat g(S) \widehat h(S) + \widehat h(S) \widehat f(S) \right) .$

Proof: Define the Boolean function ${D : \{\pm 1\}^3 \rightarrow \mathbb R}$ by

$\displaystyle D(a,b,c) = ab + bc + ca = \begin{cases} 3 & a,b,c \text{ all equal} \\ -1 & a,b,c \text{ not all equal}. \end{cases}.$

Thus paradoxical outcomes arise when ${D(f(x_\bullet), g(y_\bullet), h(z_\bullet)) = 3}$. Now, we compute that for randomly selected ${x_\bullet}$, ${y_\bullet}$, ${z_\bullet}$ that

\displaystyle \begin{aligned} \mathbf E D(f(x_\bullet), g(y_\bullet), h(z_\bullet)) &= \mathbf E \sum_S \sum_T \left( \widehat f(S) \widehat g(T) + \widehat g(S) \widehat h(T) + \widehat h(S) \widehat f(T) \right) \left( \chi_S(x_\bullet)\chi_T(y_\bullet) \right) \\ &= \sum_S \sum_T \left( \widehat f(S) \widehat g(T) + \widehat g(S) \widehat h(T) + \widehat h(S) \widehat f(T) \right) \mathbf E\left( \chi_S(x_\bullet)\chi_T(y_\bullet) \right). \end{aligned}

Now we observe that:

• If ${S \neq T}$, then ${\mathbf E \chi_S(x_\bullet) \chi_T(y_\bullet) = 0}$, since if say ${s \in S}$, ${s \notin T}$ then ${x_s}$ affects the parity of the product with 50% either way, and is independent of any other variables in the product.
• On the other hand, suppose ${S = T}$. Then

$\displaystyle \chi_S(x_\bullet) \chi_T(y_\bullet) = \prod_{s \in S} x_sy_s.$

Note that ${x_sy_s}$ is equal to ${1}$ with probability ${\frac13}$ and ${-1}$ with probability ${\frac23}$ (since ${(x_s, y_s, z_s)}$ is uniform from ${3!=6}$ choices, which we can enumerate). From this an inductive calculation on ${|S|}$ gives that

$\displaystyle \prod_{s \in S} x_sy_s = \begin{cases} +1 & \text{ with probability } \frac{1}{2}(1+(-1/3)^{|S|}) \\ -1 & \text{ with probability } \frac{1}{2}(1-(-1/3)^{|S|}). \end{cases}$

Thus

$\displaystyle \mathbf E \left( \prod_{s \in S} x_sy_s \right) = \left( -\frac13 \right)^{|S|}.$

Piecing this altogether, we now have that

$\displaystyle \mathbf E D(f(x_\bullet), g(y_\bullet), h(z_\bullet)) = \left( \widehat f(S) \widehat g(T) + \widehat g(S) \widehat h(T) + \widehat h(S) \widehat f(T) \right) \left( -\frac13 \right)^{|S|}.$

Then, we obtain that

\displaystyle \begin{aligned} &\mathbf E \frac14 \left( 1 + D(f(x_\bullet), g(y_\bullet), h(z_\bullet)) \right) \\ =& \frac14 + \frac14\sum_S \left( \widehat f(S) \widehat g(T) + \widehat g(S) \widehat h(T) + \widehat h(S) \widehat f(T) \right) \widehat f(S)^2 \left( -\frac13 \right)^{|S|}. \end{aligned}

Comparing this with the definition of ${D}$ gives the desired result. $\Box$

Now for the proof of the main theorem. We see that

$\displaystyle 1 = \sum_{S \subseteq \{1, \dots, n\}} -\left( -\frac13 \right)^{\left\lvert S \right\rvert} \left( \widehat f(S) \widehat g(S) + \widehat g(S) \widehat h(S) + \widehat h(S) \widehat f(S) \right).$

But now we can just use weak inequalities. We have ${\widehat f(\varnothing) = \mathbf E f = 0}$ and similarly for ${\widehat g}$ and ${\widehat h}$, so we restrict attention to ${|S| \ge 1}$. We then combine the famous inequality ${|ab+bc+ca| \le a^2+b^2+c^2}$ (which is true across all real numbers) to deduce that

\displaystyle \begin{aligned} 1 &= \sum_{S \subseteq \{1, \dots, n\}} -\left( -\frac13 \right)^{\left\lvert S \right\rvert} \left( \widehat f(S) \widehat g(S) + \widehat g(S) \widehat h(S) + \widehat h(S) \widehat f(S) \right) \\ &\le \sum_{S \subseteq \{1, \dots, n\}} \left( \frac13 \right)^{\left\lvert S \right\rvert} \left( \widehat f(S)^2 + \widehat g(S)^2 + \widehat h(S)^2 \right) \\ &\le \sum_{S \subseteq \{1, \dots, n\}} \left( \frac13 \right)^1 \left( \widehat f(S)^2 + \widehat g(S)^2 + \widehat h(S)^2 \right) \\ &= \frac13 (1+1+1) = 1. \end{aligned}

with the last step by Parseval. So all inequalities must be sharp, and in particular ${\widehat f}$, ${\widehat g}$, ${\widehat h}$ are supported on one-element sets, i.e. they are linear in inputs. As ${f}$, ${g}$, ${h}$ are ${\pm 1}$ valued, each ${f}$, ${g}$, ${h}$ is itself either a dictator or anti-dictator function. Since ${(f,g,h)}$ is always consistent, this implies the final result.

## 5. Pontryagin duality

In fact all the examples we have covered can be subsumed as special cases of Pontryagin duality, where we replace the domain with a general group ${G}$. In what follows, we assume ${G}$ is a locally compact abelian (LCA) group, which just means that:

• ${G}$ is a abelian topological group,
• the topology on ${G}$ is Hausdorff, and
• the topology on ${G}$ is locally compact: every point of ${G}$ has a compact neighborhood.

Notice that our previous examples fall into this category:

Example 13 (Examples of locally compact abelian groups)

• Any finite group ${Z}$ with the discrete topology is LCA.
• The circle group ${\mathbb T}$ is LCA and also in fact compact.
• The real numbers ${\mathbb R}$ are an example of an LCA group which is not compact.

### 5.1. The Pontryagin dual

The key definition is:

Definition 14

Let ${G}$ be an LCA group. Then its Pontryagin dual is the abelian group

$\displaystyle \widehat G \overset{\mathrm{def}}{=} \left\{ \text{continuous group homomorphisms } \xi : G \rightarrow \mathbb T \right\}.$

The maps ${\xi}$ are called characters. By equipping it with the compact-open topology, we make ${\widehat G}$ into an LCA group as well.

Example 15 (Examples of Pontryagin duals)

• ${\widehat{\mathbb Z} \cong \mathbb T}$.
• ${\widehat{\mathbb T} \cong \mathbb Z}$. The characters are given by ${\theta \mapsto n\theta}$ for ${n \in \mathbb Z}$.
• ${\widehat{\mathbb R} \cong \mathbb R}$. This is because a nonzero continuous homomorphism ${\mathbb R \rightarrow S^1}$ is determined by the fiber above ${1 \in S^1}$. (Covering projections, anyone?)
• ${\widehat{\mathbb Z/n\mathbb Z} \cong \mathbb Z/n\mathbb Z}$, characters ${\xi}$ being determined by the image ${\xi(1) \in \mathbb T}$.
• ${\widehat{G \times H} \cong \widehat G \times \widehat H}$.
• If ${Z}$ is a finite abelian group, then previous two examples (and structure theorem for abelian groups) imply that ${\widehat{Z} \cong Z}$, though not canonically. You may now recognize that the bilinear form ${\cdot : Z \times Z \rightarrow Z}$ is exactly a choice of isomorphism ${Z \rightarrow \widehat Z}$.
• For any group ${G}$, the dual of ${\widehat G}$ is canonically isomorphic to ${G}$, id est there is a natural isomorphism

$\displaystyle G \cong \widehat{\widehat G} \qquad \text{by} \qquad x \mapsto \left( \xi \mapsto \xi(x) \right).$

This is the Pontryagin duality theorem. (It is an analogy to the isomorphism ${(V^\vee)^\vee \cong V}$ for vector spaces ${V}$.)

### 5.2. The orthonormal basis in the compact case

Now assume ${G}$ is LCA but also compact, and thus has a unique Haar measure ${\mu}$ such that ${\mu(G) = 1}$; this lets us integrate over ${G}$. Let ${L^2(G)}$ be the space of square-integrable functions to ${\mathbb C}$, i.e.

$\displaystyle L^2(G) = \left\{ f : G \rightarrow \mathbb C \quad\text{such that}\quad \int_G |f|^2 \; d\mu < \infty \right\}.$

Thus we can equip it with the inner form

$\displaystyle \left< f,g \right> = \int_G f\overline{g} \; d\mu.$

In that case, we get all the results we wanted before:

Theorem 16 (Characters of ${\widehat G}$ forms an orthonormal basis)

Assume ${G}$ is LCA and compact. Then ${\widehat G}$ is discrete, and the characters

$\displaystyle (e_\xi)_{\xi \in \widehat G} \qquad\text{by}\qquad e_\xi(x) = e(\xi(x)) = \exp(2\pi i \xi(x))$

form an orthonormal basis of ${L^2(G)}$. Thus for each ${f \in L^2(G)}$ we have

$\displaystyle f = \sum_{\xi \in \widehat G} \widehat f(\xi) e_\xi$

where

$\displaystyle \widehat f(\xi) = \left< f, e_\xi \right> = \int_G f(x) \exp(-2\pi i \xi(x)) \; d\mu.$

The sum ${\sum_{\xi \in \widehat G}}$ makes sense since ${\widehat G}$ is discrete. In particular,

• Letting ${G = Z}$ gives “Fourier transform on finite groups”.
• The special case ${G = \mathbb Z/n\mathbb Z}$ has its own Wikipedia page.
• Letting ${G = \mathbb T}$ gives the “Fourier series” earlier.

### 5.3. The Fourier transform of the non-compact case

If ${G}$ is LCA but not compact, then Theorem~16 becomes false. On the other hand, it is still possible to define a transform, but one needs to be a little more careful. The generic example to keep in mind in what follows is ${G = \mathbb R}$.

In what follows, we fix a Haar measure ${\mu}$ for ${G}$. (This ${\mu}$ is no longer unique up to scaling, since ${\mu(G) = \infty}$.)

One considers this time the space ${L^1(G)}$ of absolutely integrable functions. Then one directly defines the Fourier transform of ${f \in L^1(G)}$ to be

$\displaystyle \widehat f(\xi) = \int_G f \overline{e_\xi} \; d\mu$

imitating the previous definitions in the absence of an inner product. This ${\widehat f}$ may not be ${L^1}$, but it is at least bounded. Then we manage to at least salvage:

Theorem 17 (Fourier inversion on ${L^1(G)}$)

Take an LCA group ${G}$ and fix a Haar measure ${\mu}$ on it. One can select a unique dual measure ${\widehat \mu}$ on ${\widehat G}$ such that if ${f \in L^1(G)}$, ${\widehat f \in L^1(\widehat G)}$, the “Fourier inversion formula”

$\displaystyle f(x) = \int_{\widehat G} \widehat f(\xi) e_\xi(x) \; d\widehat\mu.$

holds almost everywhere. It holds everywhere if ${f}$ is continuous.

Notice the extra nuance of having to select measures, because it is no longer the case that ${G}$ has a single distinguished measure.

Despite the fact that the ${e_\xi}$ no longer form an orthonormal basis, the transformed function ${\widehat f : \widehat G \rightarrow \mathbb C}$ is still often useful. In particular, they have special names for a few special ${G}$:

• If ${G = \mathbb R}$, then ${\widehat G = \mathbb R}$, and this construction gives the poorly named “(continuous) Fourier transform”.
• If ${G = \mathbb Z}$, then ${\widehat G = \mathbb T}$, and this construction gives the poorly named “DTFT..

### 5.4. Summary

In summary,

• Given any LCA group ${G}$, we can transform sufficiently nice functions on ${G}$ into functions on ${\widehat G}$.
• If ${G}$ is compact, then we have the nicest situation possible: ${L^2(G)}$ is an inner product space with ${\left< f,g \right> = \int_G f \overline{g} \; d\mu}$, and ${e_\xi}$ form an orthonormal basis across ${\widehat \xi \in \widehat G}$.
• If ${G}$ is not compact, then we no longer get an orthonormal basis or even an inner product space, but it is still possible to define the transform

$\displaystyle \widehat f : \widehat G \rightarrow \mathbb C$

for ${f \in L^1(G)}$. If ${\widehat f}$ is also in ${L^1(G)}$ we still get a “Fourier inversion formula” expressing ${f}$ in terms of ${\widehat f}$.

We summarize our various flavors of Fourier analysis for various ${G}$ in the following. In the first half ${G}$ is compact, in the second half ${G}$ is not.

$\displaystyle \begin{array}{llll} \hline \text{Name} & \text{Domain }G & \text{Dual }\widehat G & \text{Characters} \\ \hline \textbf{Binary Fourier analysis} & \{\pm1\}^n & S \subseteq \left\{ 1, \dots, n \right\} & \prod_{s \in S} x_s \\ \textbf{Fourier transform on finite groups} & Z & \xi \in \widehat Z \cong Z & e( i \xi \cdot x) \\ \textbf{Discrete Fourier transform} & \mathbb Z/n\mathbb Z & \xi \in \mathbb Z/n\mathbb Z & e(\xi x / n) \\ \textbf{Fourier series} & \mathbb T \cong [-\pi, \pi] & n \in \mathbb Z & \exp(inx) \\ \hline \textbf{Continuous Fourier transform} & \mathbb R & \xi \in \mathbb R & e(\xi x) \\ \textbf{Discrete time Fourier transform} & \mathbb Z & \xi \in \mathbb T \cong [-\pi, \pi] & \exp(i \xi n) \\ \end{array}$

You might notice that the various names are awful. This is part of the reason I got confused as a high school student: every type of Fourier series above has its own Wikipedia article. If it were up to me, we would just use the term “${G}$-Fourier transform”, and that would make everyone’s lives a lot easier.

## 6. Peter-Weyl

In fact, if ${G}$ is a Lie group, even if ${G}$ is not abelian we can still give an orthonormal basis of ${L^2(G)}$ (the square-integrable functions on ${G}$). It turns out in this case the characters are attached to complex irreducible representations of ${G}$ (and in what follows all representations are complex).

The result is given by the Peter-Weyl theorem. First, we need the following result:

Lemma 18 (Compact Lie groups have unitary reps)

Any finite-dimensional (complex) representation ${V}$ of a compact Lie group ${G}$ is unitary, meaning it can be equipped with a ${G}$-invariant inner form. Consequently, ${V}$ is completely reducible: it splits into the direct sum of irreducible representations of ${G}$.

Proof: Suppose ${B : V \times V \rightarrow \mathbb C}$ is any inner product. Equip ${G}$ with a right-invariant Haar measure ${dg}$. Then we can equip it with an “averaged” inner form

$\displaystyle \widetilde B(v,w) = \int_G B(gv, gw) \; dg.$

Then ${\widetilde B}$ is the desired ${G}$-invariant inner form. Now, the fact that ${V}$ is completely reducible follows from the fact that given a subrepresentation of ${V}$, its orthogonal complement is also a subrepresentation. $\Box$

The Peter-Weyl theorem then asserts that the finite-dimensional irreducible unitary representations essentially give an orthonormal basis for ${L^2(G)}$, in the following sense. Let ${V = (V, \rho)}$ be such a representation of ${G}$, and fix an orthonormal basis of ${e_1}$, \dots, ${e_d}$ for ${V}$ (where ${d = \dim V}$). The ${(i,j)}$th matrix coefficient for ${V}$ is then given by

$\displaystyle G \xrightarrow{\rho} \mathop{\mathrm{GL}}(V) \xrightarrow{\pi_{ij}} \mathbb C$

where ${\pi_{ij}}$ is the projection onto the ${(i,j)}$th entry of the matrix. We abbreviate ${\pi_{ij} \circ \rho}$ to ${\rho_{ij}}$. Then the theorem is:

Theorem 19 (Peter-Weyl)

Let ${G}$ be a compact Lie group. Let ${\Sigma}$ denote the (pairwise non-isomorphic) irreducible finite-dimensional unitary representations of ${G}$. Then

$\displaystyle \left\{ \sqrt{\dim V} \rho_{ij} \; \Big\vert \; (V, \rho) \in \Sigma, \text{ and } 1 \le i,j \le \dim V \right\}$

is an orthonormal basis of ${L^2(G)}$.

Strictly, I should say ${\Sigma}$ is a set of representatives of the isomorphism classes of irreducible unitary representations, one for each isomorphism class.

In the special case ${G}$ is abelian, all irreducible representations are one-dimensional. A one-dimensional representation of ${G}$ is a map ${G \hookrightarrow \mathop{\mathrm{GL}}(\mathbb C) \cong \mathbb C^\times}$, but the unitary condition implies it is actually a map ${G \hookrightarrow S^1 \cong \mathbb T}$, i.e. it is an element of ${\widehat G}$.

# Uniqueness of Solutions for DiffEq’s

Let ${V}$ be a normed finite-dimensional real vector space and let ${U \subseteq V}$ be an open set. A vector field on ${U}$ is a function ${\xi : U \rightarrow V}$. (In the words of Gaitsgory: “you should imagine a vector field as a domain, and at every point there is a little vector growing out of it.”)

The idea of a differential equation is as follows. Imagine your vector field specifies a velocity at each point. So you initially place a particle somewhere in ${U}$, and then let it move freely, guided by the arrows in the vector field. (There are plenty of good pictures online.) Intuitively, for nice ${\xi}$ it should be the case that the trajectory resulting is unique. This is the main take-away; the proof itself is just for completeness.

This is a so-called differential equation:

Definition 1

Let ${\gamma : (-\varepsilon, \varepsilon) \rightarrow U}$ be a continuous path. We say ${\gamma}$ is a solution to the differential equation defined by ${\xi}$ if for each ${t \in (-\varepsilon, \varepsilon)}$ we have

$\displaystyle \gamma'(t) = \xi(\gamma(t)).$

Example 2 (Examples of DE’s)

Let ${U = V = \mathbb R}$.

1. Consider the vector field ${\xi(x) = 1}$. Then the solutions ${\gamma}$ are just ${\gamma(t) = t+c}$.
2. Consider the vector field ${\xi(x) = x}$. Then ${\gamma}$ is a solution exactly when ${\gamma'(t) = \gamma(t)}$. It’s well-known that ${\gamma(t) = c\exp(t)}$.

Of course, you may be used to seeing differential equations which are time-dependent: i.e. something like ${\gamma'(t) = t}$, for example. In fact, you can hack this to fit in the current model using the idea that time is itself just a dimension. Suppose we want to model ${\gamma'(t) = F(\gamma(t), t)}$. Then we instead consider

$\displaystyle \xi : V \times \mathbb R \rightarrow V \times \mathbb R \qquad\text{by}\qquad \xi(v, t) = (F(v,t), 1)$

and solve the resulting differential equation over ${V \times \mathbb R}$. This does exactly what we want. Geometrically, this means making time into another dimension and imagining that our particle moves at a “constant speed through time”.

The task is then mainly about finding which conditions guarantee that our differential equation behaves nicely. The answer turns out to be:

Definition 3

The vector field ${\xi : U \rightarrow V}$ satisfies the Lipschitz condition if

$\displaystyle \left\lVert \xi(x')-\xi(x'') \right\rVert \le \Lambda \left\lVert x'-x'' \right\rVert$

holds identically for some fixed constant ${\Lambda}$.

Note that continuously differentiable implies Lipschitz.

Theorem 4 (Picard-Lindelöf)

Let ${V}$ be a finite-dimensional real vector space, and let ${\xi}$ be a vector field on a domain ${U \subseteq V}$ which satisfies the Lipschitz condition.

Then for every ${x_0 \in U}$ there exists ${(-\varepsilon,\varepsilon)}$ and ${\gamma : (-\varepsilon,\varepsilon) \rightarrow U}$ such that ${\gamma'(t) = \xi(\gamma(t))}$ and ${\gamma(0) = x_0}$. Moreover, if ${\gamma_1}$ and ${\gamma_2}$ are two solutions and ${\gamma_1(t) = \gamma_2(t)}$ for some ${t}$, then ${\gamma_1 = \gamma_2}$.

In fact, Peano’s existence theorem says that if we replace Lipschitz continuity with just continuity, then ${\gamma}$ exists but need not be unique. For example:

Example 5 (Counterexample if ${\xi}$ is not differentiable)

Let ${U = V = \mathbb R}$ and consider ${\xi(x) = x^{\frac23}}$, with ${x_0 = 0}$. Then ${\gamma(t) = 0}$ and ${\gamma(t) = \left( t/3 \right)^3}$ are both solutions to the differential equation

$\displaystyle \gamma'(t) = \gamma(t)^{\frac 23}.$

Now, for the proof of the main theorem. The main idea is the following result (sometimes called the contraction principle).

Lemma 6 (Banach Fixed-Point Theorem)

Let ${(X,d)}$ be a complete metric space. Let ${f : X \rightarrow X}$ be a map such that ${d(f(x_1), f(x_2)) < \frac{1}{2} d(x_1, x_2)}$ for any ${x_1, x_2 \in X}$. Then ${f}$ has a unique fixed point.

For the proof of the main theorem, we are given ${x_0 \in V}$. Let ${X}$ be the metric space of continuous functions from ${(-\varepsilon, \varepsilon)}$ to the complete metric space ${\overline{B}(x_0, r)}$ which is the closed ball of radius ${r}$ centered at ${x_0}$. (Here ${r > 0}$ can be arbitrary, so long as it stays in ${U}$.) It turns out that ${X}$ is itself a complete metric space when equipped with the sup norm

$\displaystyle d(f, g) = \sup_{t \in (-\varepsilon, \varepsilon)} \left\lVert f(t)-g(t) \right\rVert.$

This is well-defined since ${\overline{B}(x_0, r)}$ is compact.

We wish to use the Banach theorem on ${X}$, so we’ll rig a function ${\Phi : X \rightarrow X}$ with the property that its fixed points are solutions to the differential equation. Define it by, for every ${\gamma \in X}$,

$\displaystyle \Phi(\gamma) : t \mapsto x_0 + \int_0^t \xi(\gamma(s)) \; ds.$

This function is contrived so that ${(\Phi\gamma)(0) = x_0}$ and ${\Phi\gamma}$ is both continuous and differentiable. By the Fundamental Theorem of Calculus, the derivative is exhibited by

$\displaystyle (\Phi\gamma)'(t) = \left( \int_0^t \xi(\gamma(s)) \; ds \right)' = \xi(\gamma(t)).$

In particular, fixed points correspond exactly to solutions to our differential equation.

A priori this output has signature ${\Phi\gamma : (-\varepsilon,\varepsilon) \rightarrow V}$, so we need to check that ${\Phi\gamma(t) \in \overline{B}(x_0, r)}$. We can check that

\displaystyle \begin{aligned} \left\lVert (\Phi\gamma)(t) - x_0 \right\rVert &=\left\lVert \int_0^t \xi(\gamma(s)) \; ds \right\rVert \\ &\le \int_0^t \left\lVert \xi(\gamma(s)) \; ds \right\rVert \\ &\le t \max_{s \in [0,t]} \left\lVert \xi\gamma(s) \right\rVert \\ &< \varepsilon \cdot A \end{aligned}

where ${A = \max_{x \in \overline{B}(x_0,r)} \left\lVert \xi(x) \right\rVert}$; we have ${A < \infty}$ since ${\overline{B}(x_0,r)}$ is compact. Hence by selecting ${\varepsilon < r/A}$, the above is bounded by ${r}$, so ${\Phi\gamma}$ indeed maps into ${\overline{B}(x_0, r)}$. (Note that at this point we have not used the Lipschitz condition, only that ${\xi}$ is continuous.)

It remains to show that ${\Phi}$ is contracting. Write

\displaystyle \begin{aligned} \left\lVert (\Phi\gamma_1)(t) - (\Phi\gamma_2)(t) \right\rVert &= \left\lVert \int_{s \in [0,t]} \left( \xi(\gamma_1(s))-\xi(\gamma_2(s)) \right) \right\rVert \\ &= \int_{s \in [0,t]} \left\lVert \xi(\gamma_1(s))-\xi(\gamma_2(s)) \right\rVert \\ &\le t\Lambda \sup_{s \in [0,t]} \left\lVert \gamma_1(s)-\gamma_2(s) \right\rVert \\ &< \varepsilon\Lambda \sup_{s \in [0,t]} \left\lVert \gamma_1(s)-\gamma_2(s) \right\rVert \\ &= \varepsilon\Lambda d(\gamma_1, \gamma_2) . \end{aligned}

Hence once again for ${\varepsilon}$ sufficiently small we get ${\varepsilon\Lambda \le \frac{1}{2}}$. Since the above holds identically for ${t}$, this implies

$\displaystyle d(\Phi\gamma_1, \Phi\gamma_2) \le \frac{1}{2} d(\gamma_1, \gamma_2)$

as needed.

This is a cleaned-up version of a portion of a lecture from Math 55b in Spring 2015, instructed by Dennis Gaitsgory.