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.



fourier analytic proofs
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.


