Inverse Fourier Transform of:
$$\mathfrak{F}^{-1} \left \{ e^{-\frac{x^2}{2}}{\frac{sinx}{x}} \right \} $$
by using convolution theorem.
Since Fourier Transform convolution turns into multiplication – same property holds also for inverse. So, I am thinking that:
$$\mathfrak{F}^{-1} \left \{ e^{-\frac{x^2}{2}}{\frac{sinx}{x}} \right \}= \mathfrak{F}^{-1} \left \{ e^{-\frac{x^2}{2}} \right \} \mathfrak{F}^{-1} \left \{{\frac{sinx}{x}} \right \}$$
because of linearity.
Then first expression it is Gaussian and also its inverse is the same – so should I find only for second expression $\frac{sinx}{x}$ its Inverse Fourier Transform?
Best Answer
No, you need to actually evaluate the convolution of the inverse transforms, which looks like
$$\begin{align}\sqrt{\frac{\pi}{2}} \int_{-1}^1 dk' \, e^{-(k-k')^2/2} &= \sqrt{ \pi} \int_{(k-1)/\sqrt{2}}^{(k+1)/\sqrt{2}} du \, e^{-u^2} \\ &= \frac{\pi}{2} \left [\operatorname{erf}{\left (\frac{k+1}{\sqrt{2}}\right )} - \operatorname{erf}{\left (\frac{k-1}{\sqrt{2}} \right )} \right ] \end{align}$$