I'm hoping someone can help me with what a Fourier transform problem.

I seek the Fourier transform of $f(t)$, where:

$$f(t) = a\left(1+\mathrm{erf}\left(\frac{\ln(t)-u_1}{\sigma_1\sqrt{2}}\right)\right)\left(1+\mathrm{erf}\left(\frac{ln(t)-u_2}{\sigma_2\sqrt{2}}\right)\right)+b$$

where $a$, $b$, $u_1$, $u_2$, $\sigma_1$, $\sigma_2$ are all constants.

I think this should have an analytical solution.

The constants $a$ and $b$ should be easy to deal with, and one ought to be able to use the convolution theorem to solve for the multiplication of the two functions.

The Fourier transforms of erf$(t)$ is well known, but I get stuck pretty fast in dealing with the $\mathrm{erf}(1+\ln(t))$ form.

Any advice much appreciated!


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