Fourier transform of $f(a+b t+c t^2)$

Question

I would like to compute the Fourier transform of:

$$f(a+b t+c t^2)$$

for arbitrary $f$. I am aware of how to work out the Fourier transform for $f(a+bt)$. I have already looked at the Laplace transform of $f(t^2)$ post to use it as inspiration. But it stops short of what I really need.

I also had a look at A Spectral Analysis of Function Composition and Its Implications for Sampling in Direct Volume Visualization, by Steven Bergner but I am not sure I can use it or if there isn't a simpler way.

Thanks in advance.

Marc