Solution
= Solution
With the paper's <Fourier transform> convention, the <Fourier inversion theorem> gives
$$
\boxed{f(x)=\frac1{2\pi}\int_{-\infty}^{\infty}e^{ikx}\widetilde f(k)\,dk.}
$$
This holds at continuity points under the usual inversion hypotheses; at a jump with one-sided limits it gives their mean. The factor $1/(2\pi)$ and the sign change in the exponential are part of this normalization.