Uniform approximation of sgnx by polynomials and entire functions

In 1877, E. I. Zolotarev [19, 2] found an explicit expression, in terms of elliptic functions, of the rational function of given degree m which is uniformly closest to sgn (x) on the union of two intervals [−1,−a] ∪ [a, 1]. This result was subject to many generalizations, and it has applications in electric engineering. Surprisingly, to the best of our knowledge, the similar problem for polynomials was not solved yet, so we investigate it in this paper. For comparison, we mention here the results on the uniform approximation of |x|α, α > 0 on [−1, 1]. Polynomial approximation was studied by S. Bernstein [3, 4] who found that for the error Em(α) of the best approximation by polynomials of degree m the following limit exists: