(a) Consider the periodic function

$$f(x)=5+2 \cos \left(2 \pi x-\frac{\pi}{2}\right)+3 \cos (4 \pi x)$$

on the interval $[0,1]$. The $N$-point discrete Fourier transform of $f$ is defined by

$$F_{N}(n)=\frac{1}{N} \sum_{k=0}^{N-1} f_{k} \omega_{N}^{-n k}, \quad n=0,1, \ldots, N-1$$

where $\omega_{N}=e^{2 \pi i / N}$ and $f_{k}=f(k / N)$. Compute $F_{4}(n), n=0, \ldots, 3$, using the Fast Fourier Transform (FFT). Compare your result with what you get by computing $F_{4}(n)$ directly from $(*)$. Carefully explain all your computations.

(b) Now let $f$ be any analytic function on $\mathbb{R}$ that is periodic with period 1 . The continuous Fourier transform of $f$ is defined by

$$\hat{f}_{n}=\int_{0}^{1} f(\tau) e^{-2 \pi i n \tau} d \tau, \quad n \in \mathbb{Z}$$

Use integration by parts to show that the Fourier coefficients $\hat{f}_{n}$ decay spectrally.

Explain what it means for the discrete Fourier transform of $f$ to approximate the continuous Fourier transform with spectral accuracy. Prove that it does so.

What can you say about the behaviour of $F_{N}(N-n)$ as $N \rightarrow \infty$ for fixed $n$ ?