Consider the solution of the two-point boundary value problem

$$(2-\sin \pi x) u^{\prime \prime}+u=1, \quad-1 \leqslant x \leqslant 1$$

with periodic boundary conditions at $x=-1$ and $x=1$. Construct explicitly the linear algebraic system that arises from the application of a spectral method to the above equation.

The Fourier coefficients of $u$ are defined by

$$\hat{u}_{n}=\frac{1}{2} \int_{-1}^{1} u(\tau) e^{-i \pi n \tau} d \tau$$

Prove that the computation of the Fourier coefficients for the truncated system with $-N / 2+1 \leqslant n \leqslant N / 2$ (where $N$ is an even and positive integer, and assuming that $\hat{u}_{n}=0$ outside this range of $n$ ) reduces to the solution of a tridiagonal system of algebraic equations, which you should specify.

Explain the term convergence with spectral speed and justify its validity for the derived approximation of $u$.