Subjects: Fourier Analysis
Links: Convergence of Fourier Series, Main definitions for Fourier Analysis

Isoperimetric inequality

Suppose that $\mathrm{\Gamma }$ is simple closed curve in ${\mathbb{R}}^2$ that is piecewise smooth, with length $\ell$. Then using Green's Theorem and Curl in R2, then we can calculate the Jordan Measure of the set $\mathrm{\Omega }$ such that $\partial \mathrm{\Omega }=\mathrm{\Gamma }$. Then $$J(\Omega) \le \frac{\ell^2}{4\pi}$$
With the equality iff $\mathrm{\Gamma }$ is a circle.

The way we do it is through, just supposing that the $\ell =\pi$, and trying to get a simplified as $J(\mathrm{\Omega })\le \pi$. Considering the arc-length parametrization $\gamma :[0,2\pi ]\to {\mathbb{R}}^2$, $\gamma (s)=(x(s),y(s))$. Then we can exploit that each $x$ and $y$ is $2\pi$-periodic.

With this $$\frac1{2\pi} \int_0^{2\pi} (x'(s)^2+ y'(s)^2), ds = 1$$and $$x(s) \sim \sum a_n e^{ins} \qquad y(s) \sim b_n e^{ins}$$thus getting that $$x'(s) \sim \sum a_nin e^{ins} \qquad y(s) \sim b_nin e^{ins}$$

Using Parseval's identity is applied and gives us that $$\sum_{n \in \Bbb Z} |n|^2(|a_n|^2+|b_n|^2 ) = 1$$
By Green's Theorem we have that $$J(\Omega) = \frac1{2}\left|\int_0^{2\pi} x(s) y'(s) - y(s)x'(s), dx \right| = \pi \left|\sum_{n \in \Bbb Z} n(a_n \overline{b_n} - b_n \overline{a_n})\right|$$Then we get that $$J(\Omega) \le \pi \sum_{n \in \Bbb Z} |n|^2(|a_n|^2+|b_n|^2 ) = \pi$$.

Weyl's equidistribution theorem

If $x$ is a real number, and let $\lfloor x\rfloor$ is the greatest integer less than or equal to $x$, and we call $\lfloor x\rfloor$ is the integer part of $x$. The fractional part of $x$ is defined by $\{x\}=x-\lfloor x\rfloor$

We can look at the quotient group $\mathbb{R}/\mathbb{Z}$, then we can consider that tag each $x+\mathbb{Z}$, with the representative $\{x\}$.

If we have a real number $\gamma \ne 0$, and we look at the sequence $\gamma ,2\gamma ,3\gamma ,\dots$ If we reduce it modulo $\mathbb{Z}$, that is we look at the sequence of fractional parts $${\gamma}, {2\gamma}, {3\gamma}, \dots$$
Some trivial stuff is:

• If $\gamma \in \mathbb{Q}$, then only finitely many numbers appearing in $\{n\gamma \}$ are distinct.
• If $\gamma \in \mathbb{R}\setminus \mathbb{Q}$, then the numbers $\{n\gamma \}$ are all distinct.

Def: A sequence of numbers $x_1,x_2,\dots ,x_n,\dots$ in $[0,1)$ is said to be equidistributed if for every interval $(a,b)\subseteq [0,1)$, $$\lim_{n \to \infty} \frac{|{1 \le k \le n \mid x_n \in (a,b)}}{n} = b-a$$
We see that a sequence being equidistributed is a stronger condition that the sequence is dense, and the converse is not true, since we can create from an enumeration of the rationals, a not equidistributed.

We can transform the problem a bit, considering ${\chi }_{(a,b)}$ where $(a,b)\subseteq [0,1)$. We extend this function to $\mathbb{R}$ by periodicity, and we still denoted this extension by ${\chi }_{(a,b)}(x)$. Then, as a consequence of the definitions, we find that $$|{1 \le k \le n \mid {k\gamma} \in (a,b)}| = \sum_{k = 1}^n \chi_{(a,b)}(n \gamma)$$Then the theorem can be reformulated as the statement that $$\frac1{N} \sum_{n = 1}^N \chi_{(a, b)}(n \gamma) \to \int_0^1\chi_{(a,b)}(x), dx \qquad N \to \infty$$This reduces the number theory to analysis.

Lemma: If $f$ is continuous and periodic of period $1$, and $\gamma$ is irrational, then $$\frac1{N} \sum_{n = 1}^N f(n\gamma) \to \int_0^1f(x), dx \qquad N \to \infty$$
Cor: If $f$ is Riemann integrable in $[0,1]$, periodic of period $1$ and $\gamma$ irrational, then $$\frac1{N} \sum_{n = 1}^N f(n\gamma) \to \int_0^1f(x), dx \qquad N \to \infty$$
With this theorem, we see that $$\frac1{N} \sum_{n = 1}^N \chi_{(a, b)}(n \gamma) \to \int_0^1\chi_{(a,b)}(x), dx \qquad N \to \infty$$Thus having $\{\gamma \},\{2\gamma \},\dots$ is an equidistributed sequence of points in $[0,1)$. Trivially, we have that the sequence $\{n\gamma \}$ is dense in $[0,1)$.

There is an Ergodic Theory interpretation of this result, that the simple dynamical system given $\rho :x\mapsto x+\gamma$ being the fractional part, then ${\rho }^n:x\mapsto x+n\gamma$, where we think that the action ${\rho }^n$ taking place at the time $t=n$.

For each Riemann integrable function $f$ on the circle, we also associate the effects of the action $\rho$, and we obtain the sequence of functions $$f(x), f(\rho(x)), f(\rho^2(x)),, \dots, f(\rho^n(x)), \dots $$
In this context, the ergodicityof the system is then the statement that the "time average" $$\lim_{n \to \infty} \frac1{n}\sum_{k = 1}^n f(\rho^k(x))$$exists for each $x$ and equals the "space average" $$\int_0
{ #1}
f(x), dx$$whenever $\gamma$ is irrational

Weyl's Criterion

A sequence of real numbers $x_1,x_2,\dots$ in $[0,1)$ is equidistributed iff for all integers $k\ne 0$ one has $$\frac1{N}\sum_{n = 1}^N e^{2\pi i k x_n} \to 0 \qquad N \to \infty$$
We can also solve The Heat Equation on the Circle