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\(H^{\varvec{\infty }}\) Interpolation and Embedding Theorems for Rational Functions

  • Anton BaranovEmail author
  • Rachid Zarouf
Article
  • 13 Downloads

Abstract

We consider a Nevanlinna–Pick interpolation problem on finite sequences of the unit disc \(\mathbb {D}\) constrained by Hardy and radial-weighted Bergman norms. We find sharp asymptotics on the corresponding interpolation constants. As another application of our techniques we prove embedding theorems for rational functions. We find that the embedding of \(H^{\infty }\) into Hardy or radial-weighted Bergman spaces in \(\mathbb {D}\) is invertible on the subset of rational functions of a given degree n whose poles are separated from the unit circle and obtain asymptotically sharp estimates of the corresponding embedding constants.

Keywords

\(H^{\infty }\) interpolation Blaschke product Model space Rational function Hardy spaces Weighted Bergman spaces 

Mathematics Subject Classification

Primary 15A60 32A36 26A33 Secondary 30D55 26C15 41A10 

Notes

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Authors and Affiliations

  1. 1.Department of Mathematics and MechanicsSaint Petersburg State UniversitySt. PetersburgRussia
  2. 2.Department of Applied MathematicsBauman Moscow State Technical UniversityMoscowRussia
  3. 3.Laboratoire Apprentissage, Didactique, Evaluation, FormationAix-Marseille UniversitéMarseille Cedex 04France

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