On interpolating functions with minimal norm

Abstract:

Let ${H^\infty }$ denote the Banach algebra of bounded analytic functions in the unit disc $\{ z:|z| < 1\}$. If f is an extreme point in the unit ball of ${H^\infty }$, there is always a Blaschke product B, whose zeros form an interpolating sequence tending to one point of the unit circle, such that ${\left \| {f + Bh} \right \|_\infty } > 1$ if $h \in {H^\infty }$ and $h \ne 0$. An application of this result to the theory of best approximation is given.