Assume \(\alpha >0\) is not an integer. In papers published in 1913 and 1938, S.~N.~Bernstein established the limit $$\Lambda _{\infty ,\alpha }^{\ast }=\lim_{n\rightarrow \infty }n^{\alpha}E_{n}[ \vert x\vert ^{\alpha };L_{\infty }[ {-}1,1]] .$$ Here \(E_{n}[ \vert x\vert ^{\alpha };L_{\infty }[ -1,1] ] \) denotes the error in best uniform approximation of \(\left\vert x\right\vert ^{\alpha }$ on $\left[ {-}1,1\right] \) by polynomials of degree \(\leq n\). Bernstein proved that \(\Lambda _{\infty ,\alpha }^{\ast} \) is itself the error in best uniform approximation of \(\left\vert x\right\vert ^{\alpha }\) by entire functions of exponential type at most 1, on the whole real line. We prove that the best approximating entire function is unique, and satisfies an alternation property. We show that the scaled polynomials of best approximation converge to this unique entire function. We derive a representation for \(\Lambda _{\alpha ,\infty }^{\ast }\), as well as its \(L_{p}\) analogue for \(1\leq p<\infty \).