2025/04/24 by Gökalp Alpan, Alpan, Gökalp, Turgay Bayraktar +3
Mathematics · #advanced mathematical theories #Advanced Banach Space Theory #Holomorphic and Operator Theory
paper · doi:10.48550/arxiv.2504.17727
We generalize the theory of Widom factors to the \mathbb Cn setting. We define Widom factors of compact subsets K⊂ \mathbb Cn associated with multivariate orthogonal polynomials and weighted Chebyshev polynomials. We show that on product subsets K=K1×⋯× Kn of \mathbb Cn, where each Kj is a non-polar compact subset of \mathbb C, these quantities have universal lower bounds which directly extend one dimensional results. Under the additional assumption that each Kj is a subset of the real line, we provide improved lower bounds for Widom factors for some weight functions w; in particular, for the case w≡ 1. Finally, we define the Mahler measure of a multivariate polynomial relative to K⊂ \mathbb Cn and obtain lower bounds for this quantity on product sets.