2025/08/19 by Jim Agler, Agler, Jim, Connor Evans +6 · 1 citation
Mathematics · #Analytic and geometric function theory #Analytic function #Boundary (topology) #Class (philosophy) #Differentiable function #Function (biology) #Generalization #Hilbert space #Holomorphic and Operator Theory #Meromorphic and Entire Functions #Point (geometry) #Polydisc #Singularity #math.CV
paper · pdf · doi:10.48550/arxiv.2508.13742
published in arXiv (Cornell University) (Cornell University)
openalex publication_date 2025/08/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
We describe a generalization of the notion of a Hilbert space model of a function φ in the Schur-Agler class of the polydisc. This generalization is well adapted to the investigation of boundary behavior of φ at a mild singularity τ on the d-torus. We prove the existence of a generalized model with an enhanced continuity property at such a singularity τ. We use this result to prove the directional differentiability of a function φ in the Schur-Agler class at a singular point on the d-torus for which the Carathéodory condition holds and to calculate the corresponding directional derivative. The results of this paper extend to the polydisc \mathbbDd results of Agler, McCarthy, Tully-Doyle and Young which generalized to the bidisc the classical Julia-Wolff-Carathéodory theorem about analytic self-maps of \mathbbD.