Conjecture on convergence and continuity of the scattering transform for integrable functions

Let $f\in {\bf L}^1({{\mathbb{R}}}^d)$, and let $\overline S$ and $\overline S_J$ denote the limiting and windowed scattering transforms, respectively. Let $\overline{\mathcal P}_\infty$ be the path space equipped with the Dirac scattering metric, and let condition

$$denote the convergence condition introduced in the paper. **Scattering convergence and continuity conjecture.** Condition$$

holds for all $f\in {\bf L}^1({{\mathbb{R}}}^d)$. Moreover, if $f\in {\bf L}^1({{\mathbb{R}}}^d)$, then $\overline S f(q)$ is continuous in $\overline{\mathcal P}_\infty$ with respect to the Dirac scattering metric.

Numerical experiments indicate that the convergence property may hold for all integrable functions, and the claimed continuity is presented as analogous to continuity of the Fourier transform for integrable functions. The source does not establish these assertions, so their resolution remains open.