发表机构
Clemson University; Yerevan State University; Steklov Mathematical Institute of Russian Academy of Sciences（克莱姆森大学; 埃里温国立大学; 俄罗斯科学院斯捷克洛夫数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文探讨非线性傅里叶变换中 Baxter 定理的定量版本，通过数论构造证明否定结果，并给出肯定结果及方法局限。
AI 中文摘要
Baxter 关于单位圆上正交多项式的经典定理确立了：一个测度的线性傅里叶系数属于 ℓ¹ 当且仅当其非线性系数（即所谓的 Verblunsky 系数）也属于 ℓ¹。然而，这种等价性纯粹是定性的。我们探索了针对取值为 SU(2) 的非线性傅里叶变换（NLFT）的此类定量定理的可能表述，其中 Baxter 定理的类比最近已被获得。特别地，本文的亮点是一个针对 NLFT 的数论构造，它证明了该方向上的一些否定结果。我们还证明了一个肯定结果，并讨论了 Baxter 方法的局限性。
英文摘要
Baxter's classical theorem for orthogonal polynomials on the unit circle establishes that the linear Fourier coefficients of a measure are in $\ell^1$ if and only if the nonlinear coefficients, i.e., the so-called Verblunsky coefficients, are in $\ell^1$. 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.
Comments17 pages
