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Discrete nonlinear Fourier transforms and their inverses. (English) Zbl 07575793

Summary: We study two discretisations of the nonlinear Fourier transform of AKNS-ZS type, \(\mathscr{F}^E\) and \(\mathscr{F}^D\). Transformation \(\mathscr{F}^D\) is suitable for studying the distributions of the form \(u=\sum^{N}_{n=1}u_n\delta_{x_n}\), where \(\delta_{x_n}\) are delta functions. The poles \(x_n\) are not equidistant. The central result of the paper is the construction of recursive algorithms for inverses of these two transformations. The algorithm for \((\mathscr{F}^D)^{-1}\) is numerically more demanding than that for \((\mathscr{F}^E)^{-1}\). We describe an important symmetry property of \(\mathscr{F}^D\). It enables the reduction of the nonlinear Fourier analysis of the constant mass distributions \(u=\sum_{n=1}^Nu_c\delta_{x_n}\) for the numerically more efficient \(\mathscr{F}^E\) and its inverse.

MSC:

62-XX Statistics
39-XX Difference and functional equations
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