Is there a real-analytic way to derive the asymptotics of $\int_{-\infty}^\infty e^{ikx} e^{-k^4}\,dk$ as $|x|\to\infty$? In The Fourier Transform of the quartic Gaussian $\exp(-Ax^4)$: Hypergeometric functions, power series, steepest descent asymptotics and hyperasymptotics and extensions to $\exp(-Ax^{2n})$, Boyd derives the asymptotic of the integral
$$\mathcal{A}(x) = \int_{-\infty}^\infty e^{ikx} e^{-k^4}\,dk$$
for large $|x|$ using the method of steepest descent from complex analysis. The result is that $\mathcal{A}$ is a decaying oscillation with amplitude $\mathcal{O}\left(x^{-1/3}e^{-Cx^{4/3}}\right)$ for some $C>0$. Is there a way to derive this result using real-variable techniques?
 A: A differential equation for ${\cal A} (x) $ can be obtained as follows,
$$
\frac{d^3}{dx^3 } {\cal A} (x) = \int_{-\infty }^{\infty } dk\, (-ik^3 ) e^{ikx} e^{-k^4 } = \frac{x}{4} \int_{-\infty }^{\infty } dk\, e^{ikx} e^{-k^4 } = \frac{x}{4} {\cal A} (x)
$$
where integration by parts has been used in the second equality. The leading asymptotic behavior of this differential equation is satisfied by
$$
{\cal A} (x) = \exp (-C x^{4/3} )
$$
with
$$
C = e^{\pm i\pi /3 } \frac{3}{4^{4/3} } = \frac{3}{4^{4/3} } \left( \frac{1}{2} \pm i \frac{\sqrt{3} }{2} \right)
$$
(the third solution, with $C$ real and negative, is excluded since it diverges at large $x$). The two solutions must be suitably linearly combined to construct a real ${\cal A} (x)$. To obtain power corrections, make the ansatz
$$
{\cal A} (x) = f(x) \exp (-C x^{4/3} )
$$
yielding the following differential equation for $f$,
\begin{eqnarray*}
& f''' - 4C x^{1/3} f'' + 3\left( C^2 \frac{16}{9} x^{2/3} - C\frac{4}{9} x^{-2/3} \right) f' & \\
& + \left( x\left[ -\frac{1}{4} -C^3 \frac{64}{27} \right] + C^2 \frac{16}{9} x^{-1/3} + C\frac{8}{27} x^{-5/3} \right) f=0 &
\end{eqnarray*}
The leading order (i.e., the term in the square brackets) is of course already satisfied by the above choices of $C$, but now we can also read off the next order: Namely, the terms of order $x^{-1/3} f$ and $x^{2/3} f' $ must be brought to cancel. This requires that $f$ behaves as $f\sim x^{-1/3} $ such that the derivative yields the factor $-1/3$ that is needed to effect the cancellation. Thus, also the $x^{-1/3} $ power correction is clear. Further orders can be obtained systematically.
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