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For a compact circle set with normalized Lebesgue measure at most $\exp (-8\pi M/\epsilon )$, where $M\ge 1$ and $0<\epsilon \le 1$, the displayed bounds can be realized by a trigonometric polynomial with spectrum in a positive-frequency interval arbitrarily far from zero. Take the logarithm of a positive-real-part Schwarz integral on the unit disk of a small-mean cutoff, truncate after radial dilation, and modulate its bounded imaginary part. A prefix isolates one analytic half and thus reveals the large real part. The separation between small function norm and large Fourier partial sums drives the Kahane-Katznelson divergence theorem.

A trigonometric polynomial is real on the unit circle precisely when its finite coefficients have the displayed symmetry. To prove necessity, conjugate its values on the circle, use $\overline{z}=z^{-1}$, and compare coefficients. Multiplying a coefficient difference by $z^d$ gives an ordinary polynomial vanishing at infinitely many points, so it is zero. Symmetry also gives $p(z^{-1})=\overline{p(\overline{z})}$ for every nonzero complex $z$.

Cosine substitution is an isometry from continuous functions on $[-1,1]$ to even continuous $2\pi$-periodic functions. It satisfies $\omega (\tilde{f},\delta )\le \omega (f,\delta )$ because cosine is Lipschitz continuous with constant one. Even degree-at-most-$n$ trigonometric polynomials correspond exactly to algebraic polynomials via $T_j(\cos \theta )=\cos (j\theta )$. Averaging a periodic approximant with its reflection cannot increase its error against an even target, so the algebraic and periodic best errors are equal.

This average of Fourier partial sums reproduces every degree-at-most-$n$ trigonometric polynomial. With ${\sigma }_r=r^{-1}{\sum }_{j=0}^{r-1}{s}_j$,The bound follows because Fejér summation is a uniform-norm contraction. For $n=0$, the term involving ${\sigma }_0$ is omitted.

A trigonometric polynomial nonnegative on the unit circle has a polynomial modulus-square factor of polynomial degree at most its trigonometric order. For a nonzero trigonometric polynomial of actual order $d$, conjugate symmetry of trigonometric polynomial coefficients yields reciprocal-conjugate root pairing for $P=z^{d}p$. Even multiplicity of unit-circle roots of a nonnegative trigonometric polynomial permits pairing all $2d$ roots of a polynomial. Select representatives ${\zeta }_1,\dots ,{\zeta }_d$ and useIt follows that $p(z)=c{\prod }_i|z-{\zeta }_i{|}^2$ with $c=(-1{)}^d{p}_d/{\prod }_i\overline{{\zeta }_i}$. Evaluating away from the roots of a polynomial gives $c>0$. Thus $q(z)=\sqrt{c}{\prod }_i(z-{\zeta }_i)$ works. Constant and zero trigonometric polynomials have constant or zero factors. Different selections of roots of a polynomial and constant phases can give different factors.

For the trigonometric polynomial $F(t)={\sum }_{k=1}^N{b}_{k}e(kt)$, orthogonality of complex exponentials givesIndeed, expand each square and use ${\int }_0^{1}e(jt)dt=0$ for every nonzero integer $j$, and one for $j=0$.

Every continuous $2\pi$-periodic function admits degree-at-most-$n$ trigonometric polynomials with supremum norm error bounded by a universal constant times its modulus of continuity at scale $1/n$, for $n\ge 1$. This first-order Jackson-type estimate transfers to algebraic polynomials through cosine substitution for polynomial approximation.

If trigonometric polynomials have spectra in increasing disjoint positive-frequency intervals, the displayed summability gives a continuous uniform sum. At any cutoff its Fourier partial sum contains complete earlier blocks, at most one active prefix and no later blocks. Differences across an active prefix therefore isolate that block exactly. Fixed-size excursions occurring in arbitrarily late blocks contradict the Cauchy criterion, even when the complete blocks vanish at the point under consideration.

The normalized even kernelis nonnegative, has integral one, and is a trigonometric polynomial of degree $2(n-1)$. Its mass is concentrated on a scale $1/n$: $J_n(t)\le C\min (n,n^{-3}|t{|}^{-4})$ for $|t|\le \pi$. This controls a weighted second moment and produces the Jackson operator estimate.