Optimal Polynomial Approximants in H^p

2023/05/25 by Raymond Centner, Centner, Raymond, Raymond Cheng +3 · 1 citation
Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Analytic and geometric function theory #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical functions and polynomials #Primary 30E10 #Secondary 46E30

paper · pdf · doi:10.48550/arxiv.2305.16068

openalex publication_date 2023/05/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This work studies optimal polynomial approximants (OPAs) in the classical Hardy spaces on the unit disk, H^p (1 < p < ∞). In particular, we uncover some estimates concerning the OPAs of degree zero and one. It is also shown that if f ∈ H^p is an inner function, or if p>2 is an even integer, then the roots of the nontrivial OPA for 1/f are bounded from the origin by a distance depending only on p. For p≠ 2, these results are made possible by the novel use of a family of inequalities which are derived from a Banach space analogue of the Pythagorean theorem.

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• Metric projections, zeros of optimal polynomial approximants, and some extremal problems in Hardy spaces

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