PreprintarXiv licencearXiv
Quantitative possibilities for Baxter's theorem in nonlinear Fourier analysis
Abstract
Baxter's classical theorem for orthogonal polynomials on the unit circle establishes that the linear Fourier coefficients of a measure are in ℓ1 if and only if the nonlinear coefficients, i.e., the so-called Verblunsky coefficients, are in ℓ1. However, this equivalence is purely qualitative. We explore possible formulations of such quantitative theorems with norm and Lipschitz estimates for the SU(2)-valued nonlinear Fourier transform, for which the analog of Baxter's theorem has been recently obtained. In particular, the highlight of this paper is a number-theoretic construction for the NLFT which proves some negative results in this direction. We also prove a positive result and discuss the limitations of Baxter's method.
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BibTeX
@misc{alexis2026quantitative,
title = {{Quantitative possibilities for Baxter's theorem in nonlinear Fourier analysis}},
author = {Michel Alexis and Gevorg Mnatsakanyan and Kristina Oganesyan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01711},
eprint = {2610.01711},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01711},
}
RIS
TY - UNPB
TI - Quantitative possibilities for Baxter's theorem in nonlinear Fourier analysis
AU - Michel Alexis
AU - Gevorg Mnatsakanyan
AU - Kristina Oganesyan
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.01711
UR - https://arxiv.org/abs/2610.01711
AB - Baxter's classical theorem for orthogonal polynomials on the unit circle establishes that the linear Fourier coefficients of a measure are in ℓ^1 if and only if the nonlinear coefficients, i.e., the so-called Verblunsky coefficients, are in ℓ^1. However, this equivalence is purely qualitative. We explore possible formulations of such quantitative theorems with norm and Lipschitz estimates for the SU(2)-valued nonlinear Fourier transform, for which the analog of Baxter's theorem has been recently obtained. In particular, the highlight of this paper is a number-theoretic construction for the NLFT which proves some negative results in this direction. We also prove a positive result and discuss the limitations of Baxter's method.
ER -