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Nonlinear Fourier Analysis
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The nonlinear Fourier transform discussed in these notes is the map from the potential of a one dimensional discrete Dirac operator to the transmission and reflection coefficients thereof. Emphasis is on this being a nonlinear variant of the classical Fourier series, and on nonlinear analogues of classical analytic facts about Fourier series. These notes are a summary of a series of lectures given in 2003 at the Park City Mathematics Institute.
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Cited by 3 Pith papers
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Pointwise behavior of SU(1,1) nonlinear Fourier transform
SU(1,1) NLFT diverges pointwise on l² coefficients, implying failure of OPUC pointwise asymptotics for Szegő class measures.
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One sided orthogonal polynomials and a pointwise convergence result for $SU(2)$-valued nonlinear Fourier series
For complex measures with Szegő coefficients of opposite signs (class T−), the paper proves a Mate-Nevai-Totik universality bound and a.e. convergence of (φ*_n φ̃_n)² along lacunary sequences, a functional version of ...
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Convergence of sparse square-summable NLFT
Proves convergence of SU(1,1) and SU(2) nonlinear Fourier transforms for sparse square-summable data, yielding asymptotics for associated orthogonal polynomials on the unit circle.
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