Discuss the following:
Q: Suppose you are given the data points x = {x0,x1,x2,......xn} ^T and the function values f= {f0, f1,f2,........fn}^T, where xi > 0 for all i = 0,1,2,......n
a) For some reason, you think that h(x)= a + b*x + c*e^(arccos(x)) + d*sin(cos(T23(x))), where T23(x) is the 23rd degree Chebyshev polynomial is a great model for the data set. Find the normal equations whose solution define the best fit, in the least squares sense, for this model
b) It turns out (in this fictional setting) the basis functions {phi 0(x),phi 1(x), phi 2(x), phi 3(x)}= {1,x,e^(arccos(x)), sin(cos(T23(x))} are orthogonal with respect to summation over the nodes x against the weight functions w(x)=1. This should help you express the coefficients {a,b,c,d} in a simpler way than in (a)