For ease let us suppose that I am trying to transform the function $f(t)=\delta(a^2 -t^2)$. Therefore the Fourier transform is given by $$\frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} \delta(a^2 -t^2)\exp(-i\omega t)\,dt.$$
My question is how to evaluate $\int_{-\infty}^{\infty}\exp(-i\omega t)\,dt$, and how it relates to the delta function? I know that the Fourier transform of $\delta(t)$ is unity, but how does this change when considering $ \delta(a^2 -t^2)$ ?
Thanks very much.