An $L_p$ norm inequality related to extremal polynomials
Abdelhamid Rehouma, Herry Pripawanto Suryawan
TL;DR
This work analyzes L^p extremal approximation on the exterior of a Jordan curve E via the Szegö function $D_{E,\rho}$ and weighted Smirnov spaces. It defines the extremal constants $m_{n,E}$ through best approximation of $D_{E,\rho}/D_{E,\rho}(0)$ by polynomials in $L^{p}(G,\rho)$ and introduces the associated extremal polynomials $Q_n$, along with the auxiliary functions $J_n$ and $\Phi$. The main result proves that, as $m_{n,E}\to 0$, the integral polynomials $J_n(z)=\int_{\xi_G}^{z} Q_n^{p}(t) dt$ converge uniformly to $\Phi(z)=\int_{\xi_G}^{z} (D_{E,\rho}(t)/D_{E,\rho}(0))^{p} dt$ on compact subsets of $G$, with a quantitative bound on the convergence rate involving $\gamma_{p,q}$ and a $L^{p}$-norm of $Q_n$ in a weighted measure $\nu$. This establishes a precise link between extremal polynomial approximation in $L^{p}$ and the Szegö function under Szegö conditions, yielding convergence rates for the associated polynomials and their integral transforms.
Abstract
Let $E$ be a Jordan rectifiable curve in the complex plane and let $G$ be the bounded component of $\mathbb{C}\backslash E$. Now let $n\in \mathbb{N}$, and let $m_{n,E}$ denote the extremal constants defined by \begin{equation*}m_{n,E}=\inf \left\{ \left\Vert \dfrac{D_{E,ρ}\left( z\right) }{D_{E,ρ}\left( 0\right) }-P_{n}\left( z\right) \right\Vert_{L^{p}\left(G,ρ\right) }:P_{n}\left( ξ\right) =1\right\}\end{equation*}where $ξ$ is a fixed complex number.where $ρ$ is a weight function, $D_{E,ρ}\left( \cdot \right)$ is the so called {Szegö} function, $z\in G$, $p\geq 2.$ The infimum is taken over all polynomials $P_{n}$ of degree $n$. The $L_{p}$ associated extremal polynomials $\left\{Q_{n}\right\}_{n=1,2....}$ satisfies \begin{equation*} m_{n,E}=\left\Vert \dfrac{D_{E,ρ}\left( z\right) }{D_{E,ρ}\left(0\right) }-Q_{n}\left( z\right) \right\Vert_{L^{p}\left( G,ρ\right) }.\end{equation*} We define the functions, if $p\in $ $\mathbb{N}$ \begin{equation*}J_{n}\left( z\right) =\int_{ξ_{G}}^{z}Q_{n}^{p}\left( t\right) dt;\;z\in G\end{equation*} which are of course well defined polynomials for any $n\in \mathbb{N}$. Following the same convention , we define the function \begin{equation*}Φ\left( z\right) =\int\limits_{ξ_{G}}^{z}\left( \dfrac{D_{E,ρ}\left( t\right) }{D_{E,ρ}\left( 0\right) }\right) ^{p}dt,\end{equation*} Our main target in this paper is to show that when $m_{n,E}\longrightarrow0, $ then \begin{equation*}J_{n}\left( z\right) \text{ }\longrightarrow Φ\left( z\right)\end{equation*} uniformly on compact subsets of $G.$