Bounds for weighted Chebyshev and residual polynomials on subsets of $\mathbb{R}$

TL;DR

The paper studies bounds for weighted Chebyshev and residual polynomials on subsets of the real line by developing non-asymptotic and asymptotic estimates for Widom factors. It introduces and exploits Szegő factors $S({\mathfrak{e}},w)$ and Parreau–Widom constants $PW({\mathfrak{e}})$, and it unifies Chebyshev and residual polynomials via an $x_*$-normalized framework, with a detailed treatment of weights $w_0(x)=1/|P_m(x)|$. The main results include non-asymptotic bounds $W_n({\mathfrak{e}},w_0,x_*) \ge \frac{2S({\mathfrak{e}},w_0,x_*)}{1+e^{-2(n-m)g_{\mathfrak{e}}(x_*,\infty)}}$ and $W_n({\mathfrak{e}},w_0,x_*) \le 2S({\mathfrak{e}},w_0,x_*)\exp[ PW({\mathfrak{e}},x_*) ]$, as well as asymptotic bounds $\liminf_{n\to\infty} W_n({\mathfrak{e}},w,x_*) \ge 2S({\mathfrak{e}},w,x_*)$ and $\limsup_{n\to\infty} W_n({\mathfrak{e}},w,x_*) \le 2S({\mathfrak{e}},w,x_*)\exp[ PW({\mathfrak{e}},x_*) ]$, culminating in a Szegő-type theorem tying boundedness/positivity of Widom factors to the finiteness of the logarithmic integral of the weight against the equilibrium measure. These results extend classical Szegő theory to weighted and residual polynomials on Parreau–Widom real sets and have potential implications for Krylov subspace and related extremal problems.

Abstract

We give upper and lower bounds for weighted Chebyshev and residual polynomials on subsets of the real line. As an application, we prove a Szegő-type theorem in the setting of Parreau--Widom sets.

Theorems & Definitions (24)

• Theorem 1.1
• Theorem 2.1: Alternation Theorem
• proof
• Corollary 2.2
• proof
• Theorem 2.3
• proof
• Theorem 2.4
• proof
• Corollary 2.5