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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 $$\left(g*\hat g\right)(x)=\mathcal F[\hat g(k) g(k)](x).$$

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  • $\begingroup$ Convolution theorem allows you to convert convolutions in time domain into pointwise products in frequency domain. Then, use the fact that Fourier transform of the Fourier transform gives the original function but with negated input. $\endgroup$
    – TravorLZH
    Nov 23, 2020 at 10:07
  • $\begingroup$ @TravorLiu, the resulting equation is just as difficult to evaluate $\endgroup$
    – Cameron
    Nov 23, 2020 at 12:53

1 Answer 1

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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.

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