Q1: Consider the function $f: \mathbb{R} \rightarrow \mathbb{R}$ given by $f(x) = 1 + \cos(2x)$ as a $2 \pi$-periodic function. What is $\mathcal{F}_1(f)$ in this case?

A1: $\mathcal{F}_1(f)(x) = 1$

A2: $\mathcal{F}_1(f)(x) = \cos(2x)$

A3: $\mathcal{F}_1(f)(x) = 1+\cos(2x)$

A4: $\mathcal{F}_1(f)(x) = \cos(x)$

Q2: Consider the function $f: (-\pi, \pi] \rightarrow \mathbb{R}$ given by $f(x) = x^2$ as a $2 \pi$-periodic function. What is not correct for the Fourier series and its coefficients $a_k$, $b_k$ for this case?

A1: $a_1 = 4$

A2: $b_k = 0$ for all $k \in \mathbb{N}$.

A3: $\mathcal{F}_0(f)(x) = \frac{1}{3} \pi^2 $

A4: $a_k = \frac{4}{k^2} (-1)^k$