Extremal Polynomials and Entire Functions of Exponential Type

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.