{"paper":{"title":"Orthogonal polynomials, reproducing kernels, and zeros of optimal approximants","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Alan A. Sola, Catherine B\\'en\\'eteau, Constanze Liaw, Daniel Seco, Dmitry Khavinson","submitted_at":"2015-09-16T04:04:57Z","abstract_excerpt":"We study connections between orthogonal polynomials, reproducing kernel functions, and polynomials $p$ minimizing Dirichlet-type norms $\\|pf-1\\|_{\\alpha}$ for a given function $f$. For $\\alpha\\in [0,1]$ (which includes the Hardy and Dirichlet spaces of the disk) and general $f$, we show that such extremal polynomials are non-vanishing in the closed unit disk. For negative $\\alpha$, the weighted Bergman space case, the extremal polynomials are non-vanishing on a disk of strictly smaller radius, and zeros can move inside the unit disk. We also explain how $\\mathrm{dist}_{D_{\\alpha}}(1,f\\cdot \\ma"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.04807","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"}