Uniform approximation of sgn x by polynomials and entire functions

In 1877, E. I. Zolotarev [19, 2] found an explicit expression, in terms ofelliptic functions, of the rational function of given degree mwhich is uni-formly closest to sgn(x) on the union of two intervals [−1,−a] ∪[a,1]. Thisresult was subject to many generalizations, and it has applications in electricengineering.Surprisingly, to the best of our knowledge, the similar problem for poly-nomials was not solved yet, so we investigate it in this paper.For comparison, we mention here the results on the uniform approxima-tion of |x|