2025/07/07 by Michel Alexis, Alexis, Michel, Gevorg Mnatsakanyan +3
Mathematics · #30 #42 #Analysis of PDEs (math.AP) #Approximation Theory and Sequence Spaces #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Holomorphic and Operator Theory #Mathematical functions and polynomials
paper · pdf · doi:10.48550/arxiv.2507.05124
openalex publication_date 2025/07/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 (an ^* +bn)(an - bn ^*) of the partial SU(2)-valued nonlinear Fourier series (an, bn) under the assumption that the nonlinear Fourier series (a,b) itself satisfies both ‖b‖_L∞ (\mathbbT) < 2- \frac 1 2 and a^* is outer.