Let f:R R be continuous and 21-periodic, then we can represent it in terms of its Fourier series, f(x) = fxeikx KEZ We are interested in approximating the integral I:= f(x) dx L by the trapezoidal rule N 200 In := f(na/N). 2N n=-N+1 In the following tasks you may interchange limits freely without rigorous justification, in particular summation and integration. (a) Prove that I = 21fo N = { (b) Prove that eikan/N S 2N, ke 2NZ, 0, otherwise. n=-N+1 and use this to deduce that IN = 270 E EK ke2NZ REMARK: k 2NZ means that k = 2Nm for some integer m. (c) Deduce from (a, b) that In -- ||5270 If kl. ke2NZ\{0} (d) Suppose that (1) IKI S Me-alkl; or (i) lfxl SC|k|-P. In each case deduce from (c) a sharp (up to constants) estimate on the error |In - 11. HINT: If n(x) is non-negative and monotonically decreasing then k=k+1 n(k)