Remez-Type Inequality for Smooth Functions

TL;DR

This paper extends the classical Remez inequality to smooth functions by incorporating geometric measures of subsets, providing explicit bounds for function extrapolation based on the geometry of the set.

Contribution

It introduces a Remez-type inequality for smooth functions using geometric measures, expanding applicability beyond polynomials.

Findings

Extended Remez inequality to functions with finite smoothness.

Derived explicit lower bounds for smooth function extrapolation.

Connected geometric properties of sets to bounds in Whitney extension problems.

Abstract

The classical Remez inequality bounds the maximum of the absolute value of a polynomial $P(x)$ of degree $d$ on $[-1,1]$ through the maximum of its absolute value on any subset $Z$ of positive measure in $[-1,1]$. Similarly, in several variables the maximum of the absolute value of a polynomial $P(x)$ of degree $d$ on the unit ball $B^n \subset {\mathbb R}^n$ can be bounded through the maximum of its absolute value on any subset $Z\subset Q^n_1$ of positive $n$-measure $m_n(Z)$. In \cite{Yom} a stronger version of Remez inequality was obtained: the Lebesgue $n$-measure $m_n$ was replaced by a certain geometric quantity $\omega_{n,d}(Z)$ satisfying $\omega_{n,d}(Z)\geq m_n(Z)$ for any measurable $Z$. The quantity $\omega_{n,d}(Z)$ can be effectively estimated in terms of the metric entropy of $Z$ and it may be nonzero for discrete and even finite sets $Z$. In the present paper we extend Remez inequality to functions of finite smoothness. This is done by combining the result of \cite{Yom} with the Taylor polynomial approximation of smooth functions. As a consequence we obtain explicit lower bounds in some examples in the Whitney problem of a $C^k$-smooth extrapolation from a given set $Z$, in terms of the geometry of $Z$.