Convergence from discrete to continuous non-linear Fourier transform

In this note, we study the convergence from the discrete to the continuous non-linear Fourier transform. Relations between spectral problems and questions in complex function theory provide a new approach to the study of scattering problems and the non-linear Fourier transform (see A. Poltoratski [Ann. of Math. (2) 199 (2024), pp. 741–793]). In particular, the non-linear Fourier transform can be viewed from the perspective of spectral problems for differential operators. Results of Makarov and Poltoratski [J. Lond. Math. Soc. 108 (2023), pp. 916–977] and Poltoratski and Zhang [J. Funct. Ana. 284 (2023), 29 pp.] can be seen as results for the non-linear Fourier transform. These results are similar to some convergence problems for the discrete non-linear Fourier transform considered by Tao [An introduction to the nonlinear Fourier transform, 2002] and Tao and Thiele [Nonlinear Fourier analysis, 2012].