Asymptotics of Polynomial Interpolation and the Bernstein Constants

Abstract It is well known that the interpolation error for $$\left| x\right| ^{\alpha },\alpha >0$$ x ? , > 0 in $$L_{\infty }\left[ -1,1\right] $$ L ? - 1 by Lagrange polynomials based on zeros of Chebyshev first kind can be represented its limiting form entire functions exponential type. In this paper, we establish new asymptotic bounds these quantities when $$\alpha tends to infinity. Moreover, present some explicit constructions near best approximation }$$ norm which are process. The resulting formulas possibly indicate a general approach towards structure associated Bernstein constants.