# A question about jacobi polynomial coefficients

Suppose f(x) is a real valued function defined on [-1 1]. Let $f_k$ be the $k$th coefficient of the polynomial expansion of f in Jacobi polynomials i.e $f_k = \sum\limits_{k=0}^N f_k P_k(x)$, where $P_k$ is the kth Jacobi polynomial (the kind depends on w(x), but that doesn't matter for this question). Then I want to show that, if f is continuous, $f_k \sim O(1/\lambda_k^2)$ where $\lambda_k$ is the eigenvalue of the Sturm Liouville problem $((1 - x^2) w(x) P_k' (x))' = -\lambda_k w(x) P_k(x)$.
I started off with writing $f_k = \int\limits_{-1}^1 f(x)w(x)P_k(x)dx = -\frac{1}{\lambda_k} \int\limits_{-1}^1 f(x) ((1 - x^2) w(x) P_k' (x))'dx$. Now if I integrate by parts, the first term (of integration) vanishes because of the $1 - x^2$ term, leaving me with $\frac{1}{\lambda_k} \int\limits_{-1}^1 f'(x) (1 - x^2) w(x) P_k' (x)dx$. Now I am not sure how to show that the integral is $O(1/\lambda_k)$. Am I missing something very obvious here? Can anyone help me? Thanks a lot in advance.