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For an odd integer $q\ge 3$ and positive summable coefficients $a_k$, the Weierstrass M-test makes $f(x)={\sum }_{k\ge 0}{a}_k\cos (q^{k}x)$ continuous. Its partial sum over $q^k\le n$ is the unique best uniform approximation by degree-at-most-$n$ trigonometric polynomials, for $n\ge 1$. If $Q$ is the first omitted frequency, at $x_j=j\pi /Q$ every omitted cosine equals $(-1{)}^j$ because all frequency ratios are odd. Thus the tail attains alternating extrema equal to its coefficient sum at $2Q\ge 2n+2$ points. The trigonometric Chebyshev alternation theorem proves the formula. For $a_k=r^{-k}$, $r>1$, the largest decay exponent is $\log r/\log q$.

On the torus ${\mathbb{T}}^2$ with normalized Haar measure, choose $c_N$ so that $\int K_N=1$. The kernel is a nonnegative trigonometric polynomial. Its base $2+\cos s+\cos t$ has unique maximum four at the origin. Outside any fixed neighborhood its maximum is $q<4$, while in a smaller neighborhood of positive measure it is at least some $q^{\prime }>q$. Normalization consequently bounds the exterior kernel by $C(q/q^{\prime }{)}^N$, which tends to zero. Convolution with these kernels gives uniform trigonometric polynomial approximation to every continuous function on the torus.

For a nonzero real-on-the-circle trigonometric polynomial of actual order $d$, the polynomial $P(z)=z^{d}p(z)$ satisfies $P(z)=z^{2d}\overline{P(1/\overline{z})}$. Its constant term is the nonzero conjugate of its leading coefficient. Hence its nonzero roots of a polynomial occur in pairs $\zeta ,1/\overline{\zeta }$ with the same multiplicity of a root. The coefficient reversal and complex conjugation preserve the exponents of the paired factors. Fixed roots of a polynomial on the unit circle require nonnegativity, rather than this symmetry alone, to have even multiplicity.

Use normalized measure on the circle. If a trigonometric polynomial has frequency support $\{\pm n_j\}$ with $n_{j+1}>3n_j$, every zero sum of four signed frequencies pairs into opposite equal frequencies: the largest unpaired term would exceed the sum of the other three. Expanding the fourth moment and bounding its three possible pairings gives $\Vert p{\Vert }_4^4\le 3\Vert p{\Vert }_2^4$. The Holder inequality gives $\Vert p{\Vert }_2\le \Vert p{\Vert }_1^{1/3}\Vert p{\Vert }_4^{2/3}$ and hence the displayed estimate. Therefore an $L^1$ function with only these lacunary frequencies has square-summable coefficients: apply the estimate to its uniformly $L^1$-bounded Fejér sums and then pass to finite coefficient blocks. Finitely many extra modes do not affect the conclusion.

For real continuous $2\pi$-periodic $f$, a degree-at-most-$n$ real trigonometric polynomial $p$ is its unique best uniform approximation exactly when $f-p$ has $2n+2$ alternating extrema of magnitude $\Vert f-p{\Vert }_{\infty }$ at distinct ordered points in one period. The closing interval around the circle also counts. A strictly improving approximant would force its difference from $p$ to have at least $2n+2$ zeros, whereas a nonzero degree-$n$ trigonometric polynomial has at most $2n$.

For $0<t\le \pi$, split the sum at $\lfloor 1/t\rfloor$. The first part is bounded using $|\sin kt|\le kt$; summation by parts and the geometric bound on interval sine sums control the rest. The harmonic coefficient sum grows like $\log m$, but cancellations keep the whole trigonometric polynomial uniformly bounded. Modulating this polynomial lets a Fourier partial sum expose the large uncancelled half.

For an irrational skew shift, the averages of every continuous function $g$ converge uniformly in the starting point to $\int gd{m}_2$, with $m_2$ normalized Lebesgue measure. Nonconstant Fourier basis characters have either linear or quadratic phases. Linear phases are bounded geometric series; for quadratic phases the Van der Corput inequality for finite scalar sequences reduces to linear correlations of irrational frequency, uniformly in the starting point. Approximation by trigonometric polynomials proves the assertion and identifies every invariant Borel probability measure as $m_2$.

A sequence $(x_n)$ in $\mathbb{R}/\mathbb{Z}$ is an equidistributed sequence if and only if $N^{-1}{\sum }_{n=0}^{N-1}{e}^{2\pi im{x}_n}\to 0$ for every nonzero integer $m$. Approximation of continuous functions by trigonometric polynomials proves the criterion.