Let $u$ be a function in $L^2_{\nu}([0,\pi])$ where $\nu=2\sin(x)$
Let $\hat{u}_n$ denote the truncated Legendre series expansion of $u$ defined by \begin{equation*} \hat{u}_n:= \sum_{i=0}^{n} \frac{2i+1}{2}\langle {u}, {P_i}\rangle P_i(\cos(x)), \end{equation*} where $$\langle {u}, {P_i}\rangle = \int_0^\pi u(x)*P_i(\cos(x))d\nu(x) = \int_0^\pi u(x)*P_i(\cos(x))*(2\sin(x))d(x).$$
I'm interested in the the error approximating $u$ with $\hat{u}_n$, i.e. the difference $u-\hat{u}_n$. Since $u$ can be expanded as an infinite series of Legendre polynomials difference is simply the tail
\begin{equation} \|u-\hat{u}_n\| = \left\|\sum_{i=n+1}^{\infty}\frac{2i+1}{2}\langle {u}, {P_i}\rangle P_i(\cos(x)) \right\| \end{equation}
Since I'm interested in large $n$ I've tried using Stieltje's Theorem for the bound \begin{equation}\label{Stieltjes} |P_n(\cos(x))\leq \frac{4\sqrt{2}}{\sqrt{\pi n \sin(x)}}, \quad 0<x<\pi, n=1, 2, \ldots. \end{equation}
Using this bound in the error term I get
\begin{equation} \|u-\hat{u}_n\|\leq C*\|u\| \sum_{i=n+1}^{\infty}\frac{1}{\sqrt{n}}, \end{equation}
where $C$ is a constant. So this isn't very useful because the sum diverges.
I know there are recent results on this topic in Wang, H. and Xiang, S., 2012. On the convergence rates of Legendre approximation, but their results assume some kind of differentiability.
Can a finer error rate be achieved for a general function $u$ in $L^2_{\nu}([0,\pi])$? Thanks in advance.