On a Characterization of Spaces of Differentiable Functions

In this paper, we generalize Bernstein's theorem characterizing the space $C^k[a,b]$ by means of local approximations. The closed interval $[a,b]$ is partitioned into disjoint half-intervals on which best approximation polynomials of degree $k-1$ divided by the lengths of these half-intervals taken to the power $k$ are considered. The existence of the limits of these ratios as the lengths of the half-intervals tend to zero is a criterion for the existence of the $k$th derivative of a function. We prove the theorem in a stronger form and extend it to the spaces $W_p^k[a,b]$.