PN(x) be the trigonometric polynomial of lowest order that interpolates the

periodic array fj, j = 0,1,...,N 1 at the equidistributed nodes xj = j(2/N)

j = 0, 1, . . . , N 1, i.e

Note that PN (x) is again a trigonometric polynomial of degree N/2 and whose coefficients can be computed from those of PN(x) via the FFT. (a) Write a python code to compute a (spectral) approximation to the derivative at xj = j(2/N) for j = 0,1,...,N 1 for the corresponding periodic array fj, j = 0,1,...,N 1, using one DFT and one inverse DFT, i.e. in order N log2 N operations. (b) Test your code by comparing with your answer of the previous problem. (c) Observe the behavior of the error as N is increased to 16, and 32. Note: the contribution to the derivative from the k = N/2 node should be zero. Make sure to set the Fourier coefficient for k = N/2 of the derivative equal to zero.

N/2-1 C0 aN/2 a. COS KT 2 for x 10, 2 where 2 fj cos kxj for k 0,1,..., N/2, j or bk f, sin kr, sin kxi for k - 1... . for k=1