Dimension-free Remez Inequalities and norm designs

• URL: http://arxiv.org/abs/2310.07926v5
• Date: Wed, 20 Dec 2023 18:35:23 GMT
• Title: Dimension-free Remez Inequalities and norm designs
• Authors: Lars Becker, Ohad Klein, Joseph Slote, Alexander Volberg, Haonan Zhang
• Abstract summary: A class of domains $X$ and test sets $Y$ -- termed emphnorm -- enjoy dimension-free Remez-type estimates. We show that the supremum of $f$ does not increase by more than $mathcalO(log K)2d$ when $f$ is extended to the polytorus.
• Score: 48.5897526636987
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The classical Remez inequality bounds the supremum of a bounded-degree polynomial on an interval $X$ by its supremum on any subset $Y\subset X$ of positive Lebesgue measure. There are many multivariate generalizations of the Remez inequality, but most have constants that depend strongly on dimension. Here we show that a broad class of domains $X$ and test sets $Y$ -- termed \emph{norm designs} -- enjoy dimension-free Remez-type estimates. Instantiations of this theorem allow us for example \emph{a}) to bound the supremum of an $n$-variate degree-$d$ polynomial on the solid cube $[0,1]^n$ by its supremum on the regular grid $\{0,1/d,2/d,\ldots, 1\}^n$ independent of dimension; and \emph{b}) in the case of a degree-$d$ polynomial $f:\mathbf{Z}_K^n\to\mathbf{C}$ on the $n$-fold product of cyclic groups of order $K$, to show the supremum of $f$ does not increase by more than $\mathcal{O}(\log K)^{2d}$ when $f$ is extended to the polytorus as $f:\mathbf{T}^n\to\mathbf{C}$.

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