Analytic approximation in $L^p$ and coinvariant subspaces of the Hardy space

TL;DR

This paper extends classical results on best $H^p$-approximation of rational functions to a broader context involving inner functions, providing a new description of badly approximable functions within certain subspaces.

Contribution

It generalizes existing approximation results to coinvariant subspaces of the Hardy space associated with inner functions, offering a novel characterization of badly approximable functions.

Findings

Generalization of classical approximation results to inner function subspaces

Description of badly approximable functions in $ar heta H^p$

Enhanced understanding of $L^p$ approximation in Hardy spaces

Abstract

We generalize a classical result by A.Macintyre and W.Rogosinski on best $H^p$--approximation in $L^p$ of rational functions. For each inner function $\theta$ we give a description of $H^p$--badly approximable functions in $\bar \theta H^p$.