= Solution
The integral is linear in its first argument and conjugate-linear in its second, and <complex conjugation> of the integrand shows $\langle g,f\rangle=\overline{\langle f,g\rangle}$. Thus it is a <Hermitian form>.
If $f(z)=\sum_{j=0}^N a_jz^j$, <Fourier orthogonality> gives
$$
\langle f,f\rangle
=\sum_{j,k}a_j\overline{a_k}\frac1{2\pi}\int_0^{2\pi}e^{i(j-k)\theta}\,d\theta
=\sum_{j=0}^N|a_j|^2.
$$
This is nonnegative and vanishes only when every coefficient vanishes, that is, only for the zero <polynomial>. Hence \b[the <Hermitian form> is positive definite].
Back to article page
