We define, for every non-zero natural number $n$, the function $k_{n}$ by
$$\begin{cases} k_{n}(x) = 1 - \frac{|x|}{n} & \text{if } |x| \leqslant n \\ k_{n}(x) = 0 & \text{otherwise} \end{cases}$$
Express the Fourier transform $\hat{k}_{n}(x)$ using the function defined by
$$\varphi(x) = \begin{cases} \left(\frac{\sin x}{x}\right)^{2} & \text{if } x \neq 0 \\ 1 & \text{if } x = 0 \end{cases}$$