In this exercise, we investigate the effect of more general changes of variables on orthogonal polynomials.
(a) Prove that t = 2 s2 - 1 defines a one-to-one map from the interval 0 ≤ s ≤ 1 to the interval - 1 ≤ t ≤ 1.
(b) Let pk(t) denote the monic Legendre polynomials, which are orthogonal on - 1 ≤ t ≤ 1. Show that qk(s) = pk(2s2 - 1) defines a polynomial. Write out the cases k = 0. 1. 2. 3 explicitly.
(c) Are the polynomials qk(s) orthogonal under the L2 inner product on [0, 1]? If not, do they retain any sort of orthogonality property?