We denote by $\mathcal{C}([-1,1]^2)$ the space of continuous functions from $[-1,1]^2$ to $\mathbb{C}$ and $\mathcal{T}([-1,1]^2)$ the subspace generated by the functions $$e_{u,v} : (s,t) \mapsto e^{i\pi us}e^{i\pi vt}, \quad (u,v) \in \mathbb{Z}^2.$$ Let $a,b,c,d \in [-1,1]$ such that $a < b$ and $c < d$. Show that for every $\varepsilon < \min\left(\frac{b-a}{2}, \frac{d-c}{2}\right)$, there exists $f_\varepsilon \in \mathcal{T}([-1,1]\times[-1,1])$ satisfying the following properties:
$f_\varepsilon(s,t) \in [0,1]$ for all $(s,t) \in [-1,1]^2$,
$f_\varepsilon(s,t) \leq \varepsilon$ for $(s,t) \notin [a,b]\times[c,d]$,
$f_\varepsilon(s,t) \geq 1-\varepsilon$ for $(s,t) \in [a+\varepsilon,b-\varepsilon]\times[c+\varepsilon,d-\varepsilon]$.
We denote by $\mathcal{C}([-1,1]^2)$ the space of continuous functions from $[-1,1]^2$ to $\mathbb{C}$ and $\mathcal{T}([-1,1]^2)$ the subspace generated by the functions
$$e_{u,v} : (s,t) \mapsto e^{i\pi us}e^{i\pi vt}, \quad (u,v) \in \mathbb{Z}^2.$$
Let $a,b,c,d \in [-1,1]$ such that $a < b$ and $c < d$. Show that for every $\varepsilon < \min\left(\frac{b-a}{2}, \frac{d-c}{2}\right)$, there exists $f_\varepsilon \in \mathcal{T}([-1,1]\times[-1,1])$ satisfying the following properties:
\begin{enumerate}
\item $f_\varepsilon(s,t) \in [0,1]$ for all $(s,t) \in [-1,1]^2$,
\item $f_\varepsilon(s,t) \leq \varepsilon$ for $(s,t) \notin [a,b]\times[c,d]$,
\item $f_\varepsilon(s,t) \geq 1-\varepsilon$ for $(s,t) \in [a+\varepsilon,b-\varepsilon]\times[c+\varepsilon,d-\varepsilon]$.
\end{enumerate}