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While searching through Mathoverflow, I found out a fourier analytic proof of the Isoperimetric Inequality.Also, by google search I found a fourier analytic proof of Quadratic Reciprocity theorem.I know of the fourier analytic approach used by combinatorialists like Ben Green. But what are the other fourier analytic proofs of some of the well known classical theorems other than what I have mentioned above specially those which admit a starkly different proofs.

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Central limit theorem? –  Otis Chodosh Feb 1 '13 at 4:26
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This should be a Community Wiki question, as it is asking for a list. I've added the tag [big-list]. –  David Roberts Feb 1 '13 at 4:27
    
yes central limit theorem is a good one. i found the proof in tao's blog –  Koushik Feb 1 '13 at 8:09
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Flagged the mods to make CW. –  Benjamin Steinberg Feb 1 '13 at 17:02
    
I voted to close this question. There are simply too many Fourier-Analytic proofs to try to list them. –  Alexandre Eremenko Feb 2 '13 at 22:26
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4 Answers 4

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Hermann Weyl's delightful proof that for irrational $\alpha$ the sequence of values $k\alpha$ mod $1$, $k \in {\bf N}$, is uniformly distributed in $[0,1]$ deserves a mention. It's so simple I can summarize it here. First we check that for any nonzero $n \in {\bf Z}$ we have $$\frac{1 + e^{2\pi i n\alpha} + \cdots + e^{2\pi in(k-1)\alpha}}{k} \to 0$$ as $k \to \infty$. This is just a simple computation since the numerator is a geometric series. For $n = 0$ the displayed fraction reduces to $\frac{k}{k} = 1$. Since $\int_0^1 e^{2\pi i nx} dx = 1$ or $0$ depending on whether $n = 0$ or $n \neq 0$, it follows that $$\frac{1}{k}\sum_{j=0}^{k-1} e^{2\pi i nj\alpha} \to \int_0^1 e^{2\pi inx} dx$$ for all $n \in {\bf Z}$. Setting $x_j = j\alpha$ mod $1$ and taking linear combinations then yields $$\frac{1}{k}\sum_{j=0}^{k=1} f(x_j) \to \int_0^1 f(x) dx$$ for any trigonometric polynomial $f$, and by straightforward approximation arguments we get the same conclusion, first for any continuous function $f$ on $[0,1]$ and then for $f = \chi_{[a,b]}$. But with this $f$ the left side becomes the fraction of values $j\alpha$ mod $1$ for $0 \leq j \leq k-1$ which lie in $[a,b]$ and the right side becomes $b-a$, so this is just the statement of uniform distribution.

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What is the strikingly different proof? Obviously you mean the ergodic theoretic one, which relays on unique ergodicity of the Haar measure on the circle (wrt to infinitesimal generator of $\mathbb{T}^{1}$). But in my view, proving the unique ergodicity is almost equivalent to the proof shown above, not mentioning that Weyl was there before (the famous paper from 1913). I would mention "Hardy's theorem" (which is basically utilizing the weyl equi. criterion) of equidistribution of $n^{2}\alpha$, which have truely strikingly different proof (Furstenberg's, by skew-product construction). –  Asaf Feb 2 '13 at 17:19
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The sign of the quadratic Gauss sum $\tau$ can be obtained from the spectrum of the discrete Fourier transform $\Phi$: the trace of $\Phi$ gives $\tau$, and $\det\Phi$ distinguishes $\tau$ from $-\tau$.

Recall that for an odd prime $p$ the quadratic Gauss sum can be defined by $$ \tau = \sum_{n=0}^{p-1} \zeta^{n^2} $$ where $\zeta = e^{2\pi i / p}$. It is elementary that $|\tau|^2 = p$ and that $\tau$ is real or pure imaginary according as $p \equiv 1 \bmod 4$ or $p \equiv -1 \bmod 4$. In fact $\tau$ is always $+\sqrt p$ in the former case, and $+i\sqrt p$ in the latter, but this is notoriously tricky to prove.

One trick is to recognize $\tau$ as the trace of the discrete Fourier transform on ${\bf C}^p$, which has matrix $$ \Phi = (\zeta^{mn})_{m,n=0}^{p-1}. $$ Now $\Phi^2$ is the matrix whose $(m,n)$ entry is $p$ if $m+n \equiv 0 \bmod p$ and $0$ otherwise (this is tantamount to discrete Fourier inversion). This matrix has eiganvalues $+1$ and $-1$ with multiplicity $(p+1)/2$ and $(p-1)/2$ respectively. Hence $\Phi$ has eigenvalues $i^k \sqrt p$ ($k=0,1,2,3$) with multiplicities $m_k$ satisfying $m_0 + m_2 = (p+1)/2$ and $m_1 + m_3 = (p-1)/2$, and then $\tau = \sum_{k=0}^3 m_k i^k \sqrt p$. Since we already know $\tau$ up to sign there are only two possibilities: if $p \equiv 1 \bmod 4$ then $m_0$ or $m_2$ is $(p+3)/4$ and the other three $m_k$ are $(p-1)/4$, while if $p \equiv -1 \bmod 4$ then $m_1$ or $m_3$ is $(p-3)/4$ and the other three $m_k$ are $(p+1)/4$. We are to show that the odd man out is always $m_0$ in the former case and $m_3$ in the latter.

In each case we can decide the correct choice by computing the sign (a.k.a. argument) of $\det \Phi = p^{p/2} \prod_{k=0}^3 i^{k m_k}$. We can do this because $\Phi$ is a Vandermonde matrix, whence $\det\Phi$ has the product expansion $\prod_{0 \leq m < n < p} (\zeta^n - \zeta^m)$. Each factor $\zeta^n - \zeta^m$ is a positive real multiple of $\exp((m+n+\frac12)\pi i)$. It soon follows that $\det\Phi = i^{(1-p)/2} p^{p/2}$ (we already knew $\left|\det\Phi\right|$ because each eigenvalue has absolute value $\sqrt{p}$), and conclude as desired that $\tau = \sqrt{p}$ when $p \equiv 1 \bmod 4$ while $\tau = i\sqrt{p}$ when $p \equiv -1 \bmod 4$.

[This looks like a known but not very well-known argument that is easier to rediscover than to find in the literature. What is the original source?]

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I believe the first appearance of this proof is in the paper "Uber die Gausschen Summen" by Issai Schur. –  Aleksandar Bahat Feb 2 '13 at 21:09
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There are two very extensive monographs on this subject, more precisely, its relationship to convex geometry: by Groemer (Geometric applications of Fourier series and spherical harmonics) and Koldobsky (Fourier analysis in convex geometry).

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Let me speak about the "Triumph of Fourier" according to the words of Laurent Schwartz in his autobiography. The Fourier transformation is a handy tool to characterize regularity of functions.

Let $u$ be a distribution on some open subset $\Omega$ of $\mathbb R^n$. A consequence of the Paley-Wiener theorem is that a point $x_0$ in $\Omega$ is not in the singular support of $u$ whenever there exists a neighbordhood $U$ of $x_0$ such that $$ \forall \chi\in C_c^\infty(U), \forall N\in \mathbb N,\quad\vert\widehat{\chi u}(\xi)\vert\vert \xi\vert^N\in L^{\infty}(\mathbb R^n). $$

That notion can be refined to define the ($C^\infty$) wave-front-set, as a subset of the cotangent bundle (minus the 0 section): a point $(x_0,\xi_0)\in \Omega \times\mathbb S^{n-1}$ does not belong to the wave-front-set of $u$ whenever there exists a neighbordhood $U$ of $x_0$, a neighborhood $V$ of $\xi_0$ on the sphere such that $$ \forall \chi\in C_c^\infty(U), \forall N\in \mathbb N,\quad\vert\widehat{\chi u}(t\xi)\vert t^N\in L^{\infty}((1,+\infty)_t\times V). $$

The wave-front-set (WF) can be used to detect the various directions of singularities: for instance $$ WF(\delta_{0})=\text{\{0\}}\times (\mathbb R^n\backslash\text{\{0\}}) $$ but with $H=\mathbf 1_{(0,+\infty)}$ the Heaviside function, $H(x_1)$ in $\mathbb R^n$ is also singular at 0 but the structure of the singularity is quite different and indeed with $\Sigma=${$x\in \mathbb R^n, x_1=0$} $$ WF(H(x_1))=\Sigma\times\text{\{$0\not=\xi\in \mathbb R^n, \xi_2=\dots=\xi_n=0$\}} $$ which is the conormal bundle to $\Sigma$. The first projection of the wave-front-set is the singular support.

That definition can be extended to Sobolev regularity (spaces based on $L^2$), analytic regularity (more generally Gevrey) and is the only way to express the propagation of singularities for linear waves.

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