Please give references for the integral transform of the next kind:
$$
F\_3(f(x))(t)=\int\_{-\infty}^{\infty} \exp(Q\_3(x,t)) f(t)\,dt ,
$$
with $Q\_3(x,t)$ - a cubic polynomial of its arguments. Special cases are interesting, and of course a general case. It seems such transforms are used in PDO theory, not so? 



Fourier transform with cubic exponential
Do you mean the Airy transform?
![](https://ilorentz.org/beenakker/MO/AiryTransform.png)


A more recent reference is [The Airy transform and the associated polynomials](https://arxiv.org/abs/1002.2517) (2010).


In your notation the function $F\_3$ is the Airy transform of the Fourier transform $\hat{f}$ of $f$:


$$F\_3(x)=\int\_{-\infty}^\infty dt\,\exp\left(\tfrac{1}{3}it^3+itx\right)f(t)$$
$$\qquad=\frac{1}{2\pi}\int\_{-\infty}^\infty d\xi\,\int\_{-\infty}^\infty dt\exp\left(\tfrac{1}{3}it^3-it(\xi-x)\right)\hat{f}(\xi)$$
$$\qquad=\int\_{-\infty}^\infty d\xi\, {\rm Ai}(\xi-x)\hat{f}(\xi).$$


