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Under the displayed diagonal-sum convention, set $w(z)=(1,z^{-1},\dots ,z^{-d}{)}^T$ on the unit circle. A Hermitian matrix $M$ then givesConsequently $M⪰0$ 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.

Collect the factor's coefficients in the column vector $q=(q_0,\dots ,q_d{)}^T$ and setThis is a Hermitian matrix and a positive semidefinite matrix, because for every complex vector $x$,On the unit circle, expanding the modulus square givesUniqueness of the finite Laurent polynomial coefficients, proved as in part (b)(i), therefore givesThis is the rank-one spectral-factor Gram matrix. The matrix $M$ has matrix rank one when $q\ne 0$ and zero when $q=0$. The orientation $i-j=k$ is the PDF's convention: using $\overline{q_i}{q}_j$ instead would generally interchange $p_k$ and $p_{-k}$.