Let $P_n:[-1,1]\rightarrow\mathbb{R}$ be the $n$-th Legendre Polynomial. and, let $$a_n:=\int_{-1}^1 f(t) P_n(t)\, dt$$ for some $f:[-1,1]\rightarrow\mathbb{R}$.

Are there any good references on the decay rate of $\vert a_n \vert$? I am not familiar with this kind of problem, but I guess there must be a lot of methods.

From the following similar mathoverflow question: Reference for the exponential decay of Legendre coefficients, I found one paper. Also, I could find a book "Spherical Harmonics and Approximations on the Unit Sphere" by Atkinson. After skimming those references, it seems that showing smoothness of $f$ is one method. But the application in my mind is the case when $f=\arccos^2(t)$, which is not smooth enough.

So, I'm wondering are there any other references explaniing various methods to compute decay rate of $\vert a_n \vert$. Especially, if there are some techniques one can use for non-smooth $f$, I really want to know.