Greedy adaptive decomposition of signals based on nonlinear Fourier atoms (Q2812452)

In this paper, the authors present an adaptive decomposition of signals in terms of nonlinear Fourier atoms.NEWLINENEWLINELet \({\mathbb D} \subset {\mathbb C}\) be the open unit disc. Let \(H^2({\mathbb T})\) be the Hardy space of the complex unit circle \(\mathbb T\). The authors use the nonlinear Fourier atomic dictionary NEWLINE\[NEWLINE {\mathcal D} = \{e_{\alpha}(\exp({\mathrm i}t)) = \frac{\sqrt{1 -|\alpha|^2}}{1 - {\overline \alpha}\,\exp({\mathrm i}t)} \,:\, \alpha\in {\mathbb D}\} NEWLINE\]NEWLINE to approximate any function \(f\in H^2({\mathbb T})\) by a linear combination of finitely many \(e_{\alpha}\in {\mathcal D}\). Note that \(e_{\alpha}\in {\mathcal D}\) for \(\alpha \in \mathbb D\) are nonorthogonal functions in \(H^2(\mathbb T)\). This approximation problem is solved by an adaptive greedy algorithm, where its convergence rate is given. Numerical experiments illustrate the results.