# $H^{\infty}$ interpolation and embedding theorems for rational functions

## TL;DR

This paper investigates $H^{
abla}$ interpolation problems and embedding theorems for rational functions, providing sharp asymptotics and conditions for invertibility of embeddings between Hardy, Bergman, and rational function spaces.

## Contribution

It introduces new sharp asymptotic estimates for interpolation constants and establishes invertibility conditions for embeddings of rational functions into Hardy and Bergman spaces.

## Key findings

- Sharp asymptotics for interpolation constants.
- Invertibility conditions for embeddings of rational functions.
- Asymptotically sharp estimates of embedding constants.

## Abstract

We consider a Nevanlinna-Pick interpolation problem on finite sequences of the unit disc 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 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. Mathematics Subject Classification (2010). Primary 15A60, 32A36, 26A33; Secondary 30D55, 26C15, 41A10.