On bernstein type inequalities and a weighted chebyshev approximation problem on ellipses

Abstract

We are concerned with a classical inequality due to Bernstein which estimates the norm of polynomials on any given ellipse in terms of their norm on any smaller ellipse with the same foci. For the uniform and a certain weighted uniform norm, and for the case that the two ellipses are not “too close”, we derive sharp estimates of this type and determine the corresponding extremal polynomials. These Bernstein type inequalities are closely connected with certain constrained Chebyshev approximation problems on ellipses. We also present some new results for a weighted approximation problem of this type.