Suppose f(x) is a real valued function defined on [-1 1]. Let $\displaystyle f_k$ be the $\displaystyle k$th coefficient of the polynomial expansion of f in Jacobi polynomials i.e$\displaystyle f_k = \sum\limits_{k=0}^N f_k P_k(x) $, where$\displaystyle 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, $\displaystyle f_k \sim O(1/\lambda_k^2)$ where $\displaystyle \lambda_k$ is the eigenvalue of the Sturm Liouville problem $\displaystyle ((1 - x^2) w(x) P_k' (x))' = -\lambda_k w(x) P_k(x)$.

I started off with writing $\displaystyle 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 $\displaystyle 1 - x^2$ term, leaving me with $\displaystyle \frac{1}{\lambda_k} \int\limits_{-1}^1 f'(x) (1 - x^2) w(x) P_k' (x)dx$. Now I am not sure how to show that the integral is $\displaystyle O(1/\lambda_k) $. Am I missing something very obvious here? Can anyone help me? Thanks a lot in advance.