= Gram matrix representation of a trigonometric polynomial
{title2=$p_k=\sum_{i-j=k}M_{ij}$}
Under the displayed diagonal-sum convention, set $w(z)=(1,z^{-1},\ldots,z^{-d})^T$ on the <unit circle>. A <Hermitian matrix> $M$ then gives
$$
p(z)=w(z)^*Mw(z).
$$
Consequently $M\succeq0$ proves nonnegativity. Conversely the <Fejér–Riesz theorem> produces the <rank-one spectral-factor Gram matrix>. This gives a <semidefinite programming> representation of nonnegative trigonometric <polynomials>. The reversed convention $j-i=k$ instead uses the unconjugated monomial <vector>.
Back to article page
