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

TL;DR

This paper investigates the convergence properties of $SU(2)$-valued nonlinear Fourier series, establishing a pointwise convergence result and exploring the behavior of associated orthogonal polynomials and their zeros.

Contribution

It introduces a simplified universality convergence result for $SU(2)$ nonlinear Fourier series and relates pointwise convergence to polynomial zeros and local parameters.

Findings

Proves a convergence result for the reproducing kernel associated with $SU(2)$-valued series.

Establishes almost everywhere convergence of certain functional sequences along lacunary subsequences.

Links polynomial zero behavior to pointwise convergence of the nonlinear Fourier series.

Abstract

We elaborate on a connection between the $SU(2)$-valued nonlinear Fourier series and sequences of left and right orthogonal polynomials for complex measures on the unit circle. We show a convergence result for the associated reproducing kernel. This is a universality type result in the vein of Mate-Nevai-Totik, which turns out to be much simpler in the $SU(2)$ case than in the $SU(1,1)$ case. We then relate a.e. pointwise convergence of the product of left and right polynomials and their squares with both behavior of their zeros as well as behavior of some local parameters for these polynomials. We conclude by proving almost everywhere convergence along lacunary sequences of the functional $(a_n ^* +b_n)(a_n - b_n ^*)$ of the partial $SU(2)$-valued nonlinear Fourier series $(a_n, b_n)$ under the assumption that the nonlinear Fourier series $(a,b)$ itself satisfies both $\|b\|_{L^{\infty} (\mathbb{T})} < 2^{- \frac 1 2}$ and $a^*$ is outer.