Global Asymptotics of the Discrete Chebyshev Polynomials

TL;DR

This paper derives global asymptotic formulas for discrete Chebyshev polynomials as their degree increases, using a modified Riemann-Hilbert method, for fixed ratios of degree to parameter N.

Contribution

It introduces a modified Riemann-Hilbert approach to analyze the asymptotics of discrete Chebyshev polynomials for large degrees with fixed ratios.

Findings

Established global asymptotic formulas for tn(z, N) as n ightarrow

Applied a modified Riemann-Hilbert method for the analysis

Results valid for fixed ratio c in (0, 1)

Abstract

In this paper, we study the asymptotics of the discrete Chebyshev polynomials tn (z, N) as the degree grows to infinity. Global asymptotic formulas are obtained as n \rightarrow \infty, when the ratio of the parameters n/N = c is a constant in the interval (0, 1). Our method is based on a modified version of the Riemann-Hilbert approach first introduced by Deift and Zhou.