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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 areOne 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 $L_n(f)=f(0)-V_{n}f(0)$ 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 $|L_n(h)|\ge {\sum }_{j\ge 2n}|a_j|\gtrsim \log n/n$. 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.

The Weierstrass M-test gives a continuous periodic Weierstrass function, since $\sum a^{-k}<\infty$. For $5^m\le n<5^{m+1}$, the given best approximant leaves the tailIts supremum norm is at most the sum of its coefficients, and at $x=0$ all cosine terms equal one, attaining that bound. ConsequentlyPut $\alpha =\ln a/\ln 5$. Then $a^{-m}=(5^m{)}^{-\alpha }$ and $n/5^m\in [1,5)$, giving $E_n(g)\asymp n^{-\alpha }$, with constants depending only on $a$.

For $1<a<5$, part (a) gives $\omega (g,1/n)\le C_a{n}^{-\alpha }$. The first Jackson theorem for periodic approximation, $E_n(g)\le C\omega (g,1/n)$, supplies the matching lower bound. Hence the modulus of continuity has the precise orderAt $a=5$, the inverse theorem for trigonometric approximation gives the upper bound $C\delta \ln (1/\delta )$, and part (c) proves the matching lower bound. Thus the critical Weierstrass modulus isTo pass from scales $1/n$ to all small $\delta$, use monotonicity of the modulus of continuity and adjacent reciprocals, which are comparable to $\delta$.