Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 339 / 3

On the unit circle, $z^{-1}=\overline{z}$, soThus the trigonometric polynomial is nonnegative, with a zero at $z=-1$, andEquivalently, writing $z=e^{i\theta }$ gives $p(z)=2+2\cos \theta =4{\cos }^2(\theta /2)$. The factor has polynomial degree one as required. Multiplication of $q$ by any constant of modulus one leaves the factorization unchanged; uniqueness of $q$ is not claimed.

This is the simplest instance of the Fejér–Riesz theorem, where nonnegativity of a trigonometric polynomial on the unit circle admits a polynomial modulus-square factorization.

Since $p$ is real on the unit circle,Multiply the difference by $z^d$. It is an ordinary polynomial of polynomial degree at most $2d$ vanishing at every point of the unit circle. A nonzero polynomial has only finitely many roots of a polynomial, so all its coefficients vanish. This proves the conjugate symmetry of trigonometric polynomial coefficients:For arbitrary nonzero complex $z$, coefficient substitution now gives preciselyEquivalently $p(1/\overline{z})=\overline{p(z)}$. The complex conjugation on the right applies to the whole value, including the coefficients; it cannot simply be discarded away from the circle.

The PDF additionally assumes $p_d\ne 0$. Then $p_{-d}=\overline{p_d}\ne 0$, so the ensuing polynomial $P=z^{d}p$ has polynomial degree exactly $2d$ and nonzero constant term. These clauses and this subpart are absent from the damaged TeX, but present in the PDF.

For $z\ne 0$, the coefficient symmetry yieldsThus $P$ equals its reversed-conjugate polynomial. If $\zeta \ne 0$ is a root of a polynomial, with $P(\zeta )=0$, this identity gives $P(1/\overline{\zeta })=0$, provingThe reciprocal-conjugate root pairing also preserves the multiplicity of a root: reversal and complex conjugation take each factor associated with $\zeta$ to the corresponding factor associated with $1/\overline{\zeta }$, with the same exponent.

Since $P(0)=p_{-d}\ne 0$, no zero root of a polynomial occurs, so the reciprocal operation is always defined on the roots of a polynomial. Roots off the unit circle are paired on opposite sides of it; roots of a polynomial on the unit circle are fixed by this operation. Pairing alone does not force even multiplicity at those fixed roots of a polynomial; that extra fact uses nonnegativity.

Pair the off-circle roots of a polynomial using reciprocal-conjugate root pairing, and split each unit-circle root of a polynomial's even multiplicity equally between the two members of a pair. The fundamental theorem of algebra and the leading coefficient $p_d$ give $P(z)=p_d{\prod }_{i=1}^d(z-{\zeta }_i)(z-1/\overline{{\zeta }_i})$. None of the selected ${\zeta }_i$ is zero.

On the unit circle, the identityturns $P(z)/z^d$ intoAt a point of the unit circle outside the finite set of roots of a polynomial, the product is positive and $p(z)$ is nonzero and nonnegative. Its ratio to the product is therefore real and strictly positive. This proves $c>0$, even though the algebraic expression initially permits a complex constant. ConsequentlyThis proves the Fejér–Riesz theorem for a nonzero trigonometric polynomial of actual order $d$. A positive constant has a constant square-root factor, and the identically zero trigonometric polynomial has $q=0$; if $p_d=0$ for a specified upper order $d$, reduce to the actual order first.

Although the PDF permits assuming even multiplicity, there is a short proof of even multiplicity of unit-circle roots of a nonnegative trigonometric polynomial. The real analytic function $p(e^{i\theta })\ge 0$ cannot have a zero of odd order. Near $\zeta =e^{i{\theta }_0}$, $e^{i\theta }-\zeta$ has a simple zero, and the nonzero factor $e^{-id\theta }$ leaves the zero order of $P(e^{i\theta })$ unchanged. Hence the multiplicity of a root must be even.

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 $p_{-k}$.

For $z$ on the unit circle, use the conjugated monomial vectorThen the prescribed diagonal sums givePositive semidefiniteness provesThis is the Gram matrix representation of a trigonometric polynomial. The Hermitian condition also implies $p_{-k}=\overline{p_k}$, so its values on the unit circle are real. The conjugated monomial vector is required by the source's $i-j=k$ convention; the unconjugated vector would represent $p(z^{-1})$ instead.

In particular $p_0=\mathrm{tr}M\ge 0$. If this matrix trace is zero, all nonnegative eigenvalues vanish and $M=0$, so $p=0$. A general feasible Gram matrix need not have matrix rank one; the Fejér–Riesz theorem ensures a rank-one representative exists whenever the nonnegative trigonometric polynomial is nonzero.