Is the integral form of the polarisation identity useful for anything?

Question

It is well-known that the polarisation identity for real vector spaces is $$ \langle a,b \rangle =\frac{1}{4}\sum_{k=0}^1 (-1)^k\lVert a+(-1)^k b \rVert^2, $$ and the complex generalisation is $$ \langle a,b \rangle =\frac{1}{4}\sum_{k=0}^3 i^{-k} \lVert a+i^k b \rVert^2. $$ There are two generalisations of this: take $\omega$ a complex primitive $n$th root of unity. Then $$ \langle a,b \rangle =\frac{1}{n}\sum_{k=0}^n \omega^{-k}\lVert a+\omega^k b \rVert^2, $$ because $ \frac{1}{n}\sum_{k=0}^n \omega^{mk} = 0 $ unless $m=0$, when it's $1$. (Note we need to be careful with $n=2$, which is not enough to extract the imaginary part of a complex inner product: we instead have $$ \frac{1}{2}\sum_{k=0}^1 (-1)^{k} \lVert a+(-1)^k b \rVert^2 = \langle a,b \rangle - \langle b,a \rangle = 2\Re\langle a,b \rangle.) $$

But there's another generalisation: using Fourier series, we have $$ \langle a,b \rangle = \frac{1}{2\pi}\int_0^{2\pi} e^{-i\theta}\lVert a+e^{i\theta} b \rVert^2 \, d\theta \left( = \frac{1}{2\pi i}\int_{|z|=1} \frac{\lVert a+zb \rVert^2}{z^2} \, dz \right) $$

My question is: is this just a pretty identity, or are there situations where this is the "right"/nicest form to use in proofs requiring a polarisation identity? Obviously it adds no new mathematical content, since the $n=4$ (indeed, $n=3$) case is sufficient to determine the inner product.

Answer 1

I will write the polarization identity as:

Let $Q$ be a hermitian form. There exists a finite signed measure $\mu$ on $\{\zeta\in\mathbb{F}:\lvert\zeta\rvert=1\}$ such that $$ \langle x,y\rangle=\int_{\lvert\zeta\rvert=1}Q(x+y\,\zeta)\,\mathrm{d}\mu(\zeta) $$ is a sesquilinear form with $Q(x)=\langle x,x\rangle$ for all $x$.

(This also generalises to right-vectorspaces over $\mathbb{H}$. The situation over $\mathbb{O}$ is more complicated due to lack of associativity of multiplication but it will still work if you only demand the form to be $\mathbb{R}$-bilinear on the $x,y$ when $x\neq y$.) The exact form of $\mu$ never actually matters aesthetically in any proofs, and it is rare if ever that you actually need to compute them. Almost always the purpose of polarisation identity is to allow passing between hermitian quadratic form and sesquilinear forms.

Having said that, the usual polarization identity (the four deltas at $\{1,i,-1,-i\}$ over $\mathbb{C}$) has an advantage: you can read off the real and imaginary parts immediately. Similarly over $\mathbb{H}$ and $\mathbb{O}$ there is also a form ($4$ points $\{1,i,j,k\}$ and $8$ points $\{1,e_1,\dots,e_7\}$ respectively) that allows you to read off the real and $i,j,k$ (or $e_1,\dots,e_7$ for $\mathbb{O}$) components.