For a 2-periodic analytic function $f$, its Fourier expansion is given by the formula

$$f(x)=\sum_{n=-\infty}^{\infty} \widehat{f}_{n} e^{i \pi n x}, \quad \widehat{f}_{n}=\frac{1}{2} \int_{-1}^{1} f(t) e^{-i \pi n t} d t$$

(a) Consider the two-point boundary value problem

$$-\frac{1}{\pi^{2}}(1+2 \cos \pi x) u^{\prime \prime}+u=1+\sum_{n=1}^{\infty} \frac{2}{n^{2}+1} \cos \pi n x, \quad-1 \leqslant x \leqslant 1,$$

with periodic boundary conditions $u(-1)=u(1)$. Construct explicitly the infinite dimensional linear algebraic system that arises from the application of the Fourier spectral method to the above equation, and explain how to truncate the system to a finitedimensional one.

(b) A rectangle rule is applied to computing the integral of a 2-periodic analytic function $h$ :

$$\int_{-1}^{1} h(t) d t \approx \frac{2}{N} \sum_{k=-N / 2+1}^{N / 2} h\left(\frac{2 k}{N}\right)$$

Find an expression for the error $e_{N}(h):=\mathrm{LHS}-\mathrm{RHS}$ of $(*)$, in terms of $\widehat{h}_{n}$, and show that $e_{N}(h)$ has a spectral rate of decay as $N \rightarrow \infty$. [In the last part, you may quote a relevant theorem about $\widehat{h}_{n}$.]