Deformation of Minimal Polynomials and Approximation of Several Intervals by an Inverse Polynomial Mapping Original Research Article

In this paper we show that for a given set of l real disjoint intervals El=∪lj=1 [a2j−1, a2j] and given ε>0 there exists a real polynomial T and a set of l disjoint intervals El=∪lj=1 [ã2j−1, ã2j] with El⊇El and ‖(ã1, …, ã2l)−(a1, …, a2l)‖max<ε, such that T−1([−1, 1])=El. The statement follows by showing how to get in a constructive way by a continuous deformation procedure from a minimal polynomial on El with respect to the maximum norm a polynomial mapping of El.