- If $f$ is a bounded monotonic function on $[-\pi,\pi]$, then
$$\hat {f} (n)=O(1/|n|).$$
[Hint: One may assume that $f$ is increasing, and say $|f| \le M.$ First check that the Fourier coefficients of the charateristic function of $[a,b]$ satisfy $O(1/|n|).$ Now show that a sum of the form $$\sum_{k=1}^{N}\alpha_{k}\chi_{[a_k,a_k+1]}(x)$$ with $-\pi=a_1<a_2\ ...<a_N<a_{N+1}=\pi$ and $-M \le \alpha_{1} \le ... \le\alpha_{N}\le M$ has Fourier coefficients that are $O(1/|n|)$ uniformly in $N$. Summing by parts one gets a telescopic sum $\sum(\alpha_{k+1}-\alpha_{k})$ which can be bounded by $2M.$ Now approximate by the function of the above type.]
I don't really know how to calculate: $$\int_{-\pi}^{\pi}\sum_{k=1}^{N}\alpha_{k}\chi_{[a_k,a_k+1]}(x)e^{-inx}dx$$ Since I don't know what charateristic function is used in this case.
This characteristic function is just the function that takes value $1$ on $[a_k,a_{k+1}]$. The $\alpha_k$ are just constants. We are able to interchange sum and integral and it is a finite sum and integral.
The limits of each integral will now be $a_k$ and $a_{k+1}$. The rest is up to you! It will be an integral of the exponential function, that is where the $\frac{1}{|n|}$ comes from.
Also, the idea here is you've proved the statement for simple functions (i.e. weighted sums of characteristics), which you would then extend to monotonic functions.