We need to find a series of functions $f_m$ verifying the property $\int_{-\infty}^{+\infty}\! x^{2n} f_m\!(x) \;{\text d}x=\delta_{nm}.$ We already have found $f_0(x) = \frac{e^{-\sqrt{|x|}}\sin(\sqrt{|x|})}{|x|}$ (within a $\pi/2$ normalisation factor) using $\int_0^{+\infty}\!t^{4n+3} e^{-(1+i)t}\;{\text d}t \in\mathbb R, \quad \forall \; n \!\in\! \mathbb N$ (integrate for example along the border of the bottom right quarter of the complex plane...)
If anyone can help us find the next functions of the series, that'd be great!
Thanks!