Trigonometric Chebyshev alternation theorem ($2n+2\text{ alternating extrema}$)

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 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 $\log r/\log q$.

A standard Weierstrass function is a uniformly convergent lacunary cosine series, here parameterized by $a>1$ and an integer frequency multiplier $b>1$. Its continuity follows from the Weierstrass M-test. In the range $1<a<b$, its modulus of continuity has the fractional-power scale ${\delta }^{\log a/\log b}$ for the positive odd-frequency model; the critical case $a=b$ can have the larger scale $\delta \log (1/\delta )$. These examples distinguish small approximation error from ordinary Lipschitz continuity.

For $g(x)={\sum }_{k\ge 0}{5}^{-k}\cos (5^{k}x)$, all frequencies satisfy $5^k\equiv 1(\mathrm{mod}4)$. At $x=\pi /2$, the increment is $-{\sum }_{k\ge 0}{5}^{-k}\sin (5^{k}h)$. If $5^m\le 1/h<5^{m+1}$, the first $m+1$ terms have $0<5^{k}h\le 1$ and contribute at least $(m+1)h\sin 1$, while the remaining tail has magnitude at most $5^{-m}/4\le 5h/4$. This proves a lower modulus of continuity bound of order $h\log (1/h)$ for small $h$. The inverse theorem for trigonometric approximation supplies the matching upper bound because the positive lacunary trigonometric series has $E_n(g)\asymp n^{-1}$.