Bernstein-Walsh theory associated to convex bodies and applications to multivariate approximation theory

The Bernstein-Walsh-Siciak Theorem for analytic hypersurfaces

As a first step towards a general set-theoretic counterpart of the remarkable Bernstein-Walsh-Siciak Theorem concerning the rapidity of polynomial approximation of a holomorphic function on polynomially convex compact sets in $\mathbb{C}^n$, we prove a version of this theorem for analytic hypersurfaces.

指数位于紧凸集中的多项式及相关加权极值函数——Bernstein-Walsh-Siciak定理

Polynomials with exponents in compact convex sets and associated weighted extremal functions -- The Bernstein-Walsh-Siciak theorem

本文将Bernstein-Walsh-Siciak定理推广至指数受限于紧凸集$mS$的多项式子环，并以容许权函数下的加权一致估计替代普通一致误差估计。

Bernstein-Walsh inequalities in higher dimensions over exponential curves

Let ${\bf x}=(x_1,\dots,x_d) \in [-1,1]^d$ be linearly independent over $\mathbb Z$, set $K=\{(e^{z},e^{x_1 z},e^{x_2 z}\dots,e^{x_d z}): |z| \le 1\}.$ We prove sharp estimates for the growth of a polynomial of degree $n$, in terms of $$E_n({\bf x}):=\sup\{\|P\|_{Δ^{d+1}}:P \in \mathcal P_n(d+1), \|P\|_K \le 1\},$$ where $Δ^{d+1}$ is the unit polydisk. For all ${\bf x} \in [-1,1]^d$ with linearly independent entries, we have the lower estimate $$\log E_n({\bf x})\ge \frac{n^{d+1}}{(d-1)!(d+1)} \log n - O(n^{d+1});$$ for Diophantine $\bf x$, we have $$\log E_n({\bf x})\le \frac{ n^{d+1}}{(d-1)!…

Monge-Ampère Measures for Convex Bodies and Bernstein-Markov Type Inequalities

We use geometric methods to calculate a formula for the complex Monge-Ampère measure $(dd^cV_K)^n$, for $K \Subset \RR^n \subset \CC^n$ a convex body and $V_K$ its Siciak-Zaharjuta extremal function. Bedford and Taylor had computed this for symmetric convex bodies $K$. We apply this to show that two methods for deriving Bernstein-Markov-type inequalities, i.e., pointwise estimates of gradients of polynomials, yield the same results for all convex bodies. A key role is played by the geometric result that the extremal inscribed ellipses appearing in approximation theory are the maximal area elli…

Polynomials associated to non-convex bodies

Polynomial spaces associated to a convex body $C$ in $({\bf R}^+)^d$ have been the object of recent studies. In this work, we consider polynomial spaces associated to non-convex $C$. We develop some basic pluripotential theory including notions of $C-$extremal plurisubharmonic functions $V_{C,K}$ for $K\subset {\bf C}^d$ compact. Using this, we discuss Bernstein-Walsh type polynomial approximation results and asymptotics of random polynomials in this non-convex setting.

Polynomial estimates over exponential curves in $\mathbb C^2$

For any complex $α$ with non-zero imaginary part we show that Bernstein-Walsh type inequality holds on the piece of the curve $\{(e^z,e^{αz}) : z \in \mathbb C\}$. Our result extends a theorem of Coman-Poletsky \cite{CP10} where they considered real-valued $α$.