= Logarithmic cusp with slow trigonometric approximation
{title2=$\omega(h,\delta)\asymp\delta\log(1/\delta),\quad E_n(h)\asymp\log n/n$}
Define $h(x)=|x|\log(e\pi/|x|)$ on $[-\pi,\pi]$, set $h(0)=0$, and extend periodically. Concavity and monotonicity on $[0,\pi]$ give $\omega(h,\delta)=\delta\log(e\pi/\delta)$ for $0<\delta\le\pi$. Its cosine <Fourier coefficients> are
$$
a_j=-\frac{2}{\pi j^2}\int_0^{j\pi}\frac{1-\cos u}{u}\,du
=-\frac{2\log j}{\pi j^2}+O(j^{-2}),\qquad j\ge1.
$$
One <integration by parts> gives the formula; the remaining integral is $\log j+O(1)$ by the <Dirichlet test>. All these coefficients are nonpositive. The <de la Vallée Poussin sum> $V_n=2\sigma_{2n}-\sigma_n$ reproduces degree-at-most-$n$ <trigonometric polynomials> and has <operator norm> at most three. Hence $L_n(f)=f(0)-V_nf(0)$ has norm at most four and annihilates those polynomials. Its weights on coefficients are zero up to $n$, nonnegative thereafter, and one from $2n$ onwards, so $|L_n(h)|\ge\sum_{j\ge2n}|a_j|\gtrsim\log n/n$. Thus $E_n(h)\ge|L_n(h)|/4\gtrsim\log n/n$; the <first Jackson theorem for periodic approximation> proves the matching upper bound. This cusp and the <critical Weierstrass modulus> have the same first-modulus order but different best-approximation rates.
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