Question: Show that bn = Δf(nΔ), for n = 0, 1, ..., N - 1, are the optimal coefficients. The DFT estimate is reasonably accurate when N is large, but when N is not large there are usually better ways to estimate F(γ), as we shall see. In Figure, the real-valued function f(x) is the solid-line figure in both graphs. In the bottom graph, we see the true f(x) and a DFT estimate. The top graph is the MDFT estimator, the result of band-limited extrapolation, a technique for predicting missing Fourier coefficients that we shall discuss next.
