Construct the continuous function whose Fourier series diverges at 0.

Question

$\textbf{Problem}$ For each $n \geq 1$, there exists a continuous function $f_n$ such that $\left \vert f_n \right \vert \leq 1$ and $\left \vert S_n(f_n)(0) \right \vert \geq c' \log n$. Here, $S_n(f)(x)= (f *D_n)(x)$ where $D_n(x)$ is a drichlet kernel.

$\textbf{Attempt}$ Let \begin{equation} g_n(x) = \begin{cases} 1, \quad \textrm{if}\quad D_n(x) \geq 0, \\ -1, \quad \textrm{if} \quad D_n(x)<0. \end{cases} \end{equation} Then, we can easily check $S_n(g_n)(0) = \frac{1}{2\pi} \int_{-\pi}^{\pi} \vert D_n(y) \vert dy$. However, $g_n$ is not a continuous function! Thus, I thought the following lemma

$\textbf{Lemma}$ Suppose $f$ is integrable on $[-\pi,\pi]$ and bounded by $B$. Then, there exists a sequence $\{f_k\}_{k=1}^\infty$ of continuous functions on the circle so that
\begin{equation} \sup_{x \in [-\pi,\pi]} \vert f_k(x) \vert \leq B \quad \textrm{for all} \; k=1,2,\dots, \end{equation} and \begin{equation} \int_{-\pi}^\pi \vert f(x)- f_k(x) \vert dx \rightarrow 0 \quad \textrm{as} \; k \rightarrow \infty. \end{equation} I applied above lemma to the function $g_n$. Thus, there exists a sequence $\{h_{n_k}\}$ of continuous functions on $[-\pi,\pi]$ such that \begin{equation} \sup_{x\in [-\pi,\pi]} \vert h_{n_k}(x) \vert \leq 1 \quad \textrm{for} \; k=1,2,\dots, \quad \textrm{and} \quad \int_{-\pi}^\pi \vert g_n(x) - h_{n_k}(x) \vert dx \rightarrow 0 \quad \textrm{as} \; k\rightarrow \infty. \end{equation} Previously, I proved the following fact \begin{equation} S_n(g_n)(0) = \frac{1}{2\pi} \int_{-\pi}^\pi \vert D_n(y) \vert dy \geq c \log n. \end{equation} $\textbf{My goal}$ is to get $S_n(h_{n_k})(0)\geq c'\log n_k$. For obtaining my goal, I want to prove the difference between $S_n(g_n)(0)$ and $S_n(h_{n_k})(0)$ is very small. \begin{align} S_n(h_{n_k})(0) &= \frac{1}{2\pi} \int_{-\pi}^\pi \left(h_{n_k}(-y)-g_n(-y) \right)D_n(y) dy \; + \; \frac{1}{2\pi} \int_{-\pi}^\pi g_n(-y)D_n(y)dy \\ &=\frac{1}{2\pi} \int_{-\pi}^\pi \left(h_{n_k}(-y)-g_n(-y) \right)D_n(y) dy \; + \; S_n(g_n)(0). \end{align} However, I can't handle the first integral term in right-hand side of above equation. Any help is appreciated... Thank you!