Inner Product as Weighted Average

Let $\zeta=e^{2\pi i/n}$, where $n\geq3$. Let $||\cdot||$ be the norm induced by the complex inner product $\langle\cdot,\cdot\rangle$.

Then $$\langle x,y\rangle=\frac{1}{n}\sum_{k=1}^{n}||x+\zeta^ky||^2\zeta^k,$$ as can easily be checked.

The analogous equation $$\langle x,y\rangle=\frac{1}{2\pi}\int_{0}^{2\pi}||x+e^{i\theta}y||^2e^{i\theta}d\theta$$

also holds.

My question is: What do these mean?

It seems to suggest that $\langle x,y\rangle$ can be viewed as a weighted average of the norms of points evenly spaced on the circumference of a circle of radius $||y||$ centered at $x$; however, the significance of the weights eludes me. Are they necessary to compensate for the effect of "rotating" $y$?

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I would reserve the term "weighted average" for real, ideally non-negative, weights. When complex numbers are involved you're doing something more complicated than averaging: in real terms you're also rotating. – Qiaochu Yuan Jun 27 '12 at 1:43