(a) For a function $f=f(x, y)$ which is real analytic in $\mathbb{R}^{2}$ and 2-periodic in each variable, its Fourier expansion is given by the formula

$$f(x, y)=\sum_{m, n \in \mathbb{Z}} \widehat{f}_{m, n} e^{i \pi m x+i \pi n y}, \quad \widehat{f}_{m, n}=\frac{1}{4} \int_{-1}^{1} \int_{-1}^{1} f(t, \theta) e^{-i \pi m t-i \pi n \theta} d t d \theta$$

Derive expressions for the Fourier coefficients of partial derivatives $f_{x}, f_{y}$ and those of the product $h(x, y)=f(x, y) g(x, y)$ in terms of $\widehat{f}_{m, n}$ and $\widehat{g}_{m, n}$.

(b) Let $u(x, y)$ be the 2-periodic solution in $\mathbb{R}^{2}$ of the general second-order elliptic PDE

$$\left(a u_{x}\right)_{x}+\left(a u_{y}\right)_{y}=f$$

where $a$ and $f$ are both real analytic and 2-periodic, and $a(x, y)>0$. We impose the normalisation condition $\int_{-1}^{1} \int_{-1}^{1} u d x d y=0$ and note from the PDE $\int_{-1}^{1} \int_{-1}^{1} f d x d y=0$.

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 this system to a finite-dimensional one.

(c) Specify the truncated system for the unknowns $\left\{\widehat{u}_{m, n}\right\}$ for the case

$$a(x, y)=5+2 \cos \pi x+2 \cos \pi y$$

and prove that, for any ordering of the Fourier coefficients $\left\{\widehat{u}_{m, n}\right\}$ into one-dimensional array, the resulting system is symmetric and positive definite. [You may use the Gershgorin theorem without proof.]