I need to evaluate the integral $$ I = \int_{-\infty}^\infty \exp\Big({-i x t} + A \operatorname{sinc}(Bt)\Big)dt $$ either exactly or for any level of approximation beyond first order for $B\rightarrow 0$. Expanding the $\operatorname{sinc}$ function, the first order approximation is $$ I = \int_{-\infty}^\infty \exp\Big({-i x t} + A \Big[1-\frac{(Bt)^2}{3!}+ \cdots \Big]\Big)\,dt \approx e^A \sqrt{\frac{3! \pi}{A B^2}}\exp\Big(-\frac{3! x^2}{4 A B^2}\Big) . $$ The next order approximation would involve a convolution between this Gaussian solution and the inverse transform of a function like $e^{kt^4}$. I think this diverges.
Therefore I'm asking for advice: is there a closed form for the initial integral $I$ in terms of special functions? Or if not, is there a means to evaluate $$ J = \int_{-\infty}^\infty e^{-itx + k t^4}dt$$ in terms of special functions? Does this integral even converge? Any thoughts are appreciated !
A thought: is this related to a stable distribution? https://encyclopediaofmath.org/wiki/Stable_distribution#:~:text=The%20characteristic%20function%20of%20a%20strictly%2Dstable%20distribution%20with%20exponent,only%20be%20a%20Cauchy%20distribution.
For $$J = \int_{-\infty}^{+\infty} e^{-itx + k t^4}\,dt$$ provided that $k<0$,a CAS gives in terms of a generalized hypergeometric function $$J=\frac{2\, \Gamma \left(\frac{5}{4}\right) }{(-k)^{1/4}}\,\, _0F_2\left(;\frac{1}{2},\frac{3}{4};-\frac{x^4}{4^4 k}\right)+\frac{ \Gamma \left(-\frac{1}{4}\right)}{16 (-k)^{3/4}}x^2\,\, _0F_2\left(;\frac{5}{4},\frac{3}{2};-\frac{x^4}{4^4 k}\right)$$