关于光滑函数的Remez型不等式
Remez-Type Inequality for Smooth Functions
摘要：
经典的Remez不等式通过$[-1,1]$中任意正测度子集$Z$上的绝对值的最大值来限定$[-1,1]$上的$d$次多项式$P(x)$的绝对值的最大值.类似地，在多个变量中，单位球$B^n \subset {\mathbb R}^n$上次数为$d$的多项式$P(x)$的绝对值的最大值可以通过正$n$ -测度$m_n(Z)$的任何子集$Z\subset Q^n_1$上的其绝对值的最大值来界定。在\cite{Yom}中，得到了Remez不等式的一个更强的形式：Lebesgue $n$ -测度$m_n$被某个几何量$ω_{n,d}(Z)$代替，该几何量$ω_{n,d}(Z)$对任意可测$Z$满足$ω_{n,d}(Z)\geq m_n(Z)$.量$ω_{n,d}(Z)$可以根据$Z$的度量熵有效地估计，并且对于离散的甚至有限的集合$Z$，它可以是非零的。本文将Remez不等式推广到有限光滑函数。这是通过将\cite{Yom}的结果与光滑函数的泰勒多项式近似相结合来完成的。因此，我们得到明确的下界在一些例子中的惠特尼问题的$C^k$ -光滑外推从一个给定的集合$Z$，在$Z$的几何。
英文摘要：
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 $ω_{n,d}(Z)$ satisfying $ω_{n,d}(Z)\geq m_n(Z)$ for any measurable $Z$. The quantity $ω_{n,d}(Z)$ 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 $Z$.
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