Exercise 2.4: Weighted inner products and approximation of singular func- tions Consider the function f (t) = 1/t in the interval (0, 1]. (a) Show that f(t) is not in L2(0, 1), but that it is in the Hilbert space L2, w (0, 1), where the inner product is given by (x, y)w = x (t)y(t)w(t) dt and w (t) = t2. (b) From the set {1, t, to, to, t* ), construct a set of ON basis functions for 12, w (0, 1). These are the first five Jacobi polynomials (Abramowitz and Stegun, 1970). (c) Find a five-term approximation to 1/t with this inner product and basis. Plot the exact function and five-term approximation. Compute the error between the exact and approximate solutions using the inner product above to define a norm. This type of inner product is sometimes used in problems where the solution is known to show a singularity. As your analysis will show, polynomials can be used to get a fairly good approximation except very near the singularity