Extremal Polynomials and Entire Functions of Exponential Type

TL;DR

This paper investigates the asymptotic behavior of polynomial approximations of absolute value functions using Chebyshev zeros, deriving explicit formulas for entire functions of exponential type and exploring implications for Bernstein constants.

Contribution

It provides new asymptotic relations and explicit formulas for entire functions related to polynomial approximation, connecting to recent work on Bernstein constants.

Findings

Derived explicit formulas for entire functions of exponential type.

Established asymptotic relations for approximation errors as lphapproaches infinity.

Presented numerical results linking to Bernstein constants.

Abstract

In this paper, we discuss asymptotic relations for the approximation of $\left\vert x\right\vert ^{\alpha},\alpha>0$ in $L_{\infty}\left[ -1,1\right] $ by Lagrange interpolation polynomials based on the zeros of the Chebyshev polynomials of first kind. The limiting process reveals an entire function of exponential type for which we can present an explicit formula. As a consequence, we further deduce an asymptotic relation for the Approximation error when $\alpha\rightarrow\infty$. Finally, we present connections of our results together with some recent work of Ganzburg [5] and Lubinsky [10], by presenting numerical results, indicating a possible constructive way towards a representation for the Bernstein constants.