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Nonlinear Fourier Analysis

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arxiv 1201.5129 v1 pith:ESF3TWVD submitted 2012-01-24 math.CA

Terence Tao , Christoph Thiele This is my paper · ORCID
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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 6 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Pointwise behavior of SU(1,1) nonlinear Fourier transform

    math.CA 2026-05 unverdicted novelty 8.0 of 10

    SU(1,1) NLFT diverges pointwise on l² coefficients, implying failure of OPUC pointwise asymptotics for Szegő class measures.

  2. Direct and inverse spectral continuity for Dirac operators

    math.SP 2025-05 conditional novelty 8.0 of 10

    Half-line Dirac operators with L2 potentials admit an explicit, two-sided weighted stability estimate between potentials and their Schur spectral functions, proved via an exact Kronig-Penney model and Schur's algorithm.

  3. One sided orthogonal polynomials and a pointwise convergence result for $SU(2)$-valued nonlinear Fourier series

    math.CA 2025-07 conditional novelty 7.0 of 10

    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 ...

  4. Inverse nonlinear fast Fourier transform on SU(2) with applications to quantum signal processing

    quant-ph 2025-05 conditional novelty 7.0 of 10

    A new O(n log^2 n) divide-and-conquer algorithm computes inverse nonlinear Fourier transforms on SU(2) with proven numerical stability under an outerness condition, and applies to QSP and generalized QSP phase factors.

  5. A Solovay-Kitaev theorem for quantum signal processing

    quant-ph 2025-05 reject novelty 7.0 of 10

    A lifted Solovay-Kitaev argument proves that density of QSP ansätze in function spaces implies the existence of short approximating circuits, with examples for several QSP variants.

  6. Convergence of sparse square-summable NLFT

    math.CA 2026-06 unverdicted novelty 5.0 of 10

    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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