H ^∞ interpolation constrained by Beurling–Sobolev norms

Abstract

We consider a Nevanlinna–Pick interpolation problem on finite sequences of the unit disc, constrained by Beurling–Sobolev norms. We find sharp asymptotics of the corresponding interpolation quantities, thereby improving the known estimates. On our way we obtain a S. M. Nikolskii type inequality for rational functions whose poles lie outside of the unit disc. It shows that the embedding of the Hardy space H 2 into the Wiener algebra of absolutely convergent Fourier/Taylor series is invertible on the subset of rational functions of a given degree, whose poles remain at a given distance from the unit circle.