{"paper":{"title":"An extension of Herglotz's theorem to the quaternions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"D. Alpay, D.P. Kimsey, F. Colombo, I. Sabadini","submitted_at":"2014-03-01T12:47:51Z","abstract_excerpt":"A classical theorem of Herglotz states that a function $n\\mapsto r(n)$ from $\\mathbb Z$ into $\\mathbb C^{s\\times s}$ is positive definite if and only there exists a $\\mathbb C^{s\\times s}$-valued positive measure $d\\mu$ on $[0,2\\pi]$ such that $r(n)=\\int_0^{2\\pi}e^{int}d\\mu(t)$for $n\\in \\mathbb Z$. We prove a quaternionic analogue of this result when the function is allowed to have a number of negative squares. A key tool in the argument is the theory of slice hyperholomorphic functions, and the representation of such functions which have a positive real part in the unit ball of the quaternion"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1403.0079","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"}