= Solution
For $T>0$ the difference of <Heaviside step functions> is an interval indicator, and $v=1/2$ on $(-T,T)$ and zero elsewhere. Its <Fourier transform> is
$$
\widetilde v(\omega,T)=\frac12\int_{-T}^{T}e^{-i\omega t}\,dt
=\boxed{\frac{\sin(\omega T)}{\omega}},\qquad
\boxed{\widetilde v(0,T)=T}.
$$
The apparent singularity at $\omega=0$ is removable. The formula remains valid for real negative $T$, because the original step-function difference and the oriented integral both reverse sign; for $T=0$ both vanish. This is the <Fourier transform of an interval indicator> and produces a scaled <sinc function>.
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