This problem is from Stein's Fourier Analysis, Chapter 2 (page 66). In the first part of the question I've managed to show that:
$$ L_N := \frac{1}{2\pi}\int^{\pi}_{-\pi}|D_N(\theta)|d\theta \geq \frac{4}{\pi^2}\log(N)$$
The second part of the problem is as follows:
Prove the following consequence: For every $n\geq1$ there exists a continuous function $f_n$ such that $|f_n|\leq1$ and $|Sn(f_n)(0)|\geq c' \log n$ for some constant $c'>0$.
Hint: The function $g_n$ which is equal to $1$ when $D_n$ is positive and $-1$ when $D_n$ is negative has the desired property but is not continuous. Approximate $g_n$ in the integral norm (in the sense of Lemma 3.2) by continuous functions $h_k$ satisfying $|h_k| ≤ 1$.
The only thing I need help clarifying is why $g_n$ given in the hint satisfy the second condition ($|Sn(f_n)(0)|\geq c' \log n$).
Is it necessary to find where $D_n$ is positive/negative in order to show this? I thought so since the Fourier coefficients need to be evaluated and in order to do so we need to find the domain of integration.
Or is this an implication from the first part of the problem?