For a given $q \in(0,1]$, recall the set $\mathbb{B}_q\left(R_q\right)$ defined in Equation (11.7) as a model of soft sparsity.
(a) A related object is the weak $\ell_q$-ball with parameters $(C, \alpha)$, given by
$$
\mathbb{B}_{w(\alpha)}(C):=\left\{\left.\theta \in \mathbb{R}^p| | \theta\right|_{(j)} \leq C j^{-\alpha} \quad \text { for } j=1, \ldots, p\right\}
$$
Here $|\theta|_{(j)}$ denote the order statistics of $\theta$ in absolute value, ordered from largest to smallest (so that $|\theta|_{(1)}=\max _{j=1,2, \ldots, p}\left|\theta_j\right|$ and $|\theta|_{(p)}=\min _{j=1,2, \ldots, p}\left|\theta_j\right|$ ) For any $\alpha>1 / q$, show that there is a radius $R_q$ depending on $(C, \alpha)$ such that $\mathbb{B}_{w(\alpha)}(C) \subseteq \mathbb{B}_q\left(R_q\right)$.
(b) For a given integer $k \in\{1,2, \ldots, p\}$, the best $k$-term approximation to a vector $\theta^* \in \mathbb{R}^p$ is given by
$$
\Pi_k\left(\theta^*\right):=\arg \min _{\|\theta\|_0 \leq k}\left\|\theta-\theta^*\right\|_2^2
$$
Give a closed form expression for $\Pi_k\left(\theta^*\right)$.
(c) When $\theta^* \in \mathbb{B}_q\left(R_q\right)$ for some $q \in(0,1]$, show that the best $k$-term approximation satisfies
$$
\left\|\Pi_k\left(\theta^*\right)-\theta^*\right\|_2^2 \leq\left(R_q\right)^{2 / q}\left(\frac{1}{k}\right)^{\frac{2}{q}-1}
$$