Convolution of a function and its Fourier transform

Question

I would like to analytically evaluate the following $$ \left(g*\hat g\right)(x) $$ where I have defined $g(k)=e^{\frac i 3 (2\pi k)^3}$ and hence its Fourier transform is $$ \mathcal F[g(k)](x)=\hat g(x)=\text{Ai}(x). $$ Are there any tricks that I could use to evaluate this for $x\in\mathbb R$?
Note that $$\left(g*\hat g\right)(x)=\mathcal F[\hat g(k) g(k)](x).$$

Answer 1

The answer is of the form

$$c_0\text{Ai}(x)+c_1\text{Ai}'(x)$$

The coefficients can be found by evaluating the integrals at x=0.