Orthogonal polynomials on the unit circle

Remarks

The condition on the support makes $\ip{p}{p} > 0$ for every nonzero polynomial $p$, since a nonzero polynomial has finitely many zeros. Then $\Phi_n$ is $z^n$ minus its orthogonal projection onto the span of $1, z, \dots, z^{n-1}$, which shows existence and uniqueness. Such measures are called nontrivial.

Example: let $\mu$ be the normalized arc length $d\theta / 2\pi$, with $z = e^{i\theta}$. Then $\ip{z^n}{z^k} = \frac{1}{2\pi} \int_0^{2\pi} e^{i(n - k)\theta} \, d\theta = 0$ for $k \ne n$, so $\Phi_n(z) = z^n$ and $\norm{\Phi_n} = 1$.

Non-example: let $\mu = \delta_1$, the point mass at $1$. Its support is finite, and the polynomial $z - 1$ has norm $0$. The orthogonality conditions for $n = 1$ give $\ip{z + c}{1} = 1 + c = 0$, so $\Phi_1 = z - 1$, which is zero in $L^2(\mu)$; the construction breaks down and does not give an orthonormal sequence.