(30 points) The normalized Chebyshev polynomials on x E [-1, 1] are defined as Tn(x) = 2/T cos(n cos 1 x), for n 2 1 and To(x) = 1/7. Consider the expansion n Sn(a) = ak Tk (2 ), k=0 where {To, T1, . .. , In} is a finite set of Chebyshev polynomials on [-1, 1]. (1) Show the normalized Chebyshev polynomials are orthonormal with (Im, In) w = om,n under the inner-product ( 2, v) w = / u(x)v(x)w(x)dx, where w (x) = 71-12 2) To approximate the function f(x) E C[-1, 1], find the coefficients ak, k = 0, . .. , n so that Sn(x) minimizes the least square error E(ao, . . . , an) = / (f(x) - Sn(x))?w(x) dx, -1 where w(x) = You are required to show the derivation process