{"paper":{"title":"Facial behaviour of analytic functions on the bidisk","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Jim Agler, John E. McCarthy, N. J. Young","submitted_at":"2010-03-17T16:03:59Z","abstract_excerpt":"We prove that if $\\phi$ is an analytic function bounded by 1 on the bidisk and $\\tau$ is a point in a face of the bidisk at which $\\phi$ satisfies Caratheodory's condition then both $\\phi$ and the angular gradient $\\nabla\\phi$ exist and are constant on the face. Moreover, the class of all $\\phi$ with prescribed $\\phi(\\tau)$ and $\\nabla\\phi(\\tau)$ can be parametrized in terms of a function in the two-variable Pick class. As an application we solve an interpolation problem with nodes that lie on faces of the bidisk."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1003.3400","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}