Suppose f(x) is a real valued function defined on [-1 1]. Let f_k be the kth 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.