Suppose f(x) is a real valued function defined on [-1 1]. Let be the th coefficient of the polynomial expansion of f in Jacobi polynomials i.e , where 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, where is the eigenvalue of the Sturm Liouville problem .
I started off with writing . Now if I integrate by parts, the first term (of integration) vanishes because of the term, leaving me with . Now I am not sure how to show that the integral is . Am I missing something very obvious here? Can anyone help me? Thanks a lot in advance.