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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

Marc Lilley
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    why don't you complete the square, you'l get something like $(a_1t+b_1)^2+c_1$ and then do similar change of variables as the question you are referring to: $u=(a_1t+b_1)^2$ – Vasili Apr 19 '18 at 17:15

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