Consider the model (2.46) used in Example 2.9, xt = Xn j=0 1(j/n) cos(2tj/n) + 2(j/n) sin(2tj/n). (a) Display the model design matrix Z [see

Consider the model (2.46) used in Example 2.9, xt = Xn j=0

β1(j/n) cos(2πtj/n) + β2(j/n) sin(2πtj/n).

(a) Display the model design matrix Z [see (2.5)] for n = 4.

(b) Show numerically that the columns of Z in part

(a) satisfy part

(d) and then display (Z0 Z)−1 for this case.

(c) If x1, x2, x3, x4 are four observations, write the estimates of the four betas,

β1(0), β1(1/4), β2(1/4), β1(1/2), in terms of the observations.

(d) Verify that for any positive integer n and j, k = 0, 1, . . . , [[n/2]], where [[·]]

denotes the greatest integer function:10

(i) Except for j = 0 or j = n/2, Xn t=1 cos2(2πtj/n) = Xn t=1 sin2(2πtj/n) = n/2

.

(ii) When j = 0 or j = n/2, Xn t=1 cos2(2πtj/n) = n but Xn t=1 sin2(2πtj/n) = 0.

(iii) For j 6= k, Xn t=1 cos(2πtj/n) cos(2πtk/n) = Xn t=1 sin(2πtj/n) sin(2πtk/n) = 0.

Also, for any j and k, Xn t=1 cos(2πtj/n) sin(2πtk/n) = 0.

10 Some useful facts: 2 cos(α) = eiα + e−iα, 2i sin(α) = eiα − e−iα, and Pn t=1 zt =

z(1 − zn)/(1 − z) for z 6= 1.