- From Proposition 8.1.13, we know that the Fourier coefficients of the function fe L([0,T]; R) satisfy c-k Ck. Thus, the Fourier series satisfies = = SfCkeiwkt = co+Cewkt kЄZ + Che¹wkt kЄZ+ =co+(Ck+C) cos(wkt) + i(ck-ck) sin(wkt). kЄZ+ Show that for each ke Z+ the following holds: (Ck+Ck) cos(wkt) + i(ck - Ck) sin(wkt) = 2|ck| cos(wkt + k), == where the real and imaginary parts of ck satisfy R(ck) = |ck| cos(ok) and 3(Ck) |ck| sin(ok), respectively. In other words, we can decompose the Fourier series S[f] into a sum = S[f](t)=co+2|ck| cos(wkt + k) k=1 (8.80) of linear oscillators, each having frequency k/T, amplitude 2|ck|, and phase Фк. Proposition 8.1.13. If ƒ e L2([0,T]; R), then Ckck for each kЄ Z. we have Proof. Since f(t) = f(t) and e-iwkteikt, 1 Ck= (t)et= = f(t)edt = ck- D