Suppose we consider an $L^2$-function

$f:[0,1]\rightarrow \mathbb{R}_{\ge 0}$.

How does the property "$f$ is a.e. bounded by a rational function" translate in terms of the Fourier coefficients?

I should say: I don't really care about whether there is a $\textit{rational}$ function all too specifically. I just want to ensure that my function is everywhere locally bounded except for finitely many points, and some growth control at these exceptional points - e.g. in the form of a rational function; but other types of growth estimates would also be welcome, e.g. anything like knowing a bound of the shape

$f \le \frac{1}{x-\frac{1}{2}}$ or $f \le -log(x)$

would be great.