With attention to Wikipedia we know:
If $f$ is $2\pi$ periodic and absolutely continuous on $[0,2\pi]$, then the Fourier series of $f$ converges uniformly
Now if we show the Fourier series of $f$ like this:
\begin{align} \sum\limits_{-\infty}^{\infty}\hat{f}(n)e^{i 2\pi t \frac{n}{T}} \end{align}
Then Cauchy condition for a uniform convergence Fourier series implies: \begin{align*} \forall \epsilon > 0 \ \exists n \in \mathbb{N} : |\hat{f}(n)| < \epsilon \end{align*} which means $\hat{f}(n) \to 0$ uniformly.
Now my question is: If Fourier series of $f$ converges to $f$ uniformly, then when does $|\hat{f}(n)|$ decrease to $0$ monotonically? That means:
\begin{align} \forall n \in \mathbb{N} : |\hat{f}(n+1)| \leq |\hat{f}(n)| \end{align}