Solution
= Solution
Since $\theta_n$ is even,
$$
\widehat\theta_n(\xi)
=2\int_0^n\left(1-\frac xn\right)\cos(x\xi)\,dx.
$$
For $\xi\ne0$, integration by parts gives
$$
\widehat\theta_n(\xi)
=\frac{2(1-\cos(n\xi))}{n\xi^2}
=n\left(\frac{\sin(n\xi/2)}{n\xi/2}\right)^2.
$$
At $\xi=0$ the continuous limiting value is the area of the triangle,
$$
\widehat\theta_n(0)=\int_{-n}^n\left(1-\frac{|x|}{n}\right)dx=n.
$$
In particular, $\widehat\theta_n\geq0$.