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