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