Applications to Mathematics

We first brought up Motivations of Fourier Analysis with regards to the string and heat equations

Fourier Analysis has broad applications in other areas of mathematics

The Isoperimetric Inequality

Let $\Gamma$ denoted a closed curve in the plane which does not intersect itself with length $l$ and area $\mathcal{A}$. For a given $l$, which curve $\Gamma$ maximizes $\mathcal{A}$, if such a curve exists?

A little basic intuition tells us this is a circle, since any non-convex region could be deformed to increase its area, and this is correct but difficult to make precise

A parametrized curve $\gamma$ is a mapping $\gamma :[a,b]\to {\mathbb{R}}^2$

Definition: The image of $\gamma$ is a set of points in the plain which we call a curve and denote by $\Gamma$

$\Gamma$ is simple if it does not intersect itself, and closed if its two end-points coincide, i.e. $\gamma (s_1)\ne \gamma (s_2)$ unless $s_1=a$ and $s_2=b$

Isoperimetric Inequality: Suppose that $\Gamma$ is a simple closed curve in ${\mathbb{R}}^2$ of length $l$ and let $\mathcal{A}$ denote the area of the region enclosed by this curve. Then $\mathcal{A}\le \frac{l^2}{4\pi }$ with equality if and only if $\Gamma$ is a circle.

We can assume $\gamma$ is of class $C^1$ and assume ${\gamma }^{\prime }(s)\ne 0$, which guarantees $\Gamma$ has a well-defined tangent at each point which varies continuously as the point on the curve varies, with the parametrization inducing an orientation on $\Gamma$

Any $C^1$ bijective mapping $s:[c,d]\to [a,b]$ gives rise to another parametrization of $\Gamma$ by $\eta (t)=\gamma (s(t))$

We say $\gamma$ and $\eta$ are equivalent if $s^{\prime }(t)>0$ for all $t$, i.e. $\eta$ and $\gamma$ induce the same orientation

The length of the curve $\Gamma$ by the parametrization $\gamma (s)=(x(s),y(s))$ is $l={\int }_a^b|{\gamma }^{\prime }(s)|ds={\int }_a^b(x^{\prime }(s{)}^2+y^{\prime }(s{)}^2{)}^{1/2}ds$, and as desired, this formula is independent of the particular parametrization

$\gamma$ is called a parametrization by arc-length if $|{\gamma }^{\prime }(s)|=1$, i.e. $\gamma (s)$ travels at a constant speed, in which case $l=b-a$

Green’s Theorem admits the formula $\mathcal{A}=\frac{1}{2}|{\int }_{\Gamma }^{}(xdy-ydx)=\frac{1}{2}|{\int }_a^{b}x(s)y^{\prime }(s)-y(s)x^{\prime }(s)ds|$, which we can use with some assumptions

We’d like to show $\mathcal{A}\le \frac{l^2}{4\pi }$ with equality if and only if $\Gamma$ is a circle, however we can use the generic case $l=2\pi$ so that $\mathcal{A}\le \pi$

Let $\gamma :[0,2\pi ]\to {\mathbb{R}}^2$ with $\gamma (s)=(x(s),y(s))$ be a parametrization by arc-length of $\Gamma$
$\frac{1}{2\pi }{\int }_0^{2\pi }(x^{\prime }(s{)}^2+y^{\prime }(s{)}^2)ds=1$

$x(s)$ and $y(s)$ are $2\pi$-periodic, so we consider their Fourier series $x(s)\sim {\sum }_{}^{}{a}_n{e}^{ins}$ and $y(s)\sim {\sum }_{}^{}{b}_n{e}^{ins}$

Parseval’s identity on the above equation gives
${\sum }_{n=-\infty }^{\infty }|n{|}^2(|a_n{|}^2+|b_n{|}^2)=1$

We can then derive the following relation with another application of Parseval’s identity,
$\mathcal{A}=\frac{1}{2}|{\int }_0^{2\pi }x(s)y^{\prime }(s)-y(s)x^{\prime }(s)ds|=\pi |{\sum }_{n=-\infty }^{\infty }n(a_n{\bar{b}}_n-b_n{\bar{a}}_n)|$

$|a_n{\bar{b}}_n-b_n{\bar{a}}_n|\le 2|a_n||b_n|\le |a_n{|}^2+|b_n{|}^2$
Thus,
$\mathcal{A}\le \pi {\sum }_{n=-\infty }^{\infty }|n{|}^2(|a_n{|}^2+|b_n{|}^2)\le \pi$

When $\mathcal{A}=\pi$,
$x(s)=a_{-1}{e}^{-is}+a_0+a_1{e}^{is}$ and $y(s)=b_{-1}{e}^{-is}+b_0+b_1{e}^{is}$, since $|n|=|n{|}^2⟺|n|\le 1$

Since $x(s)$ and $y(s)$ are real-valued, $a_{-1}={\bar{a}}_1$ and $b_{-1}={\bar{b}}_1$, and we can use our equalities to derive $a_1=\frac{1}{2}{e}^{i\alpha }$ and $b_1=\frac{1}{2}{e}^{i\beta }$

$1=2|a_1\overline{b_1}-{\bar{a}}_1{b}_1|$ implies $|\sin (\alpha -\beta )|=1$, from which we can use some subtle steps to derive $x(s)=a_0+\cos (\alpha +s)$ and $y(s)=b_0\pm \sin (\alpha +s)$, which is a circle

This proves our inequality with a couple caveats. What exactly is the region enclosed by $\Gamma$ and what is the definition of its area? Hopefully this matches the theorem we used for our result. And does this result apply to curves which are merely rectifiable (but not necessarily smooth)?