Laurent Baratchart, INRIA-Sophia, France

On the rational approximation of Blaschke products

Abstract

It is easy to see that infinite Blaschke products are badly approximable by rational functions in H^∞. That is, the best approximation by a rational function of degree n to a given infinite Blaschke product in uniform norm on the circle is zero. In contrast, they are approximable in L^p norms, p < ∞, in the sense that the approximation error will tend to zero as n goes large. In this talk, we give a lower estimate on the speed of convergence in terms of the zeros of the Blaschke product. The estimate rests on a min-max principle of Courant type for singular numbers of Hankel operators, which also involves the geometry of Blaschke product in its proof.