Interpolating by functions from model subspaces in $H^1$

TL;DR

This paper characterizes when interpolation problems in a specific subspace of the Hardy space $H^1$ can be solved using functions from that subspace, based on Blaschke products and their zeros.

Contribution

It provides a new characterization of interpolating sequences for a model subspace of $H^1$ defined by Blaschke products.

Findings

Identifies conditions on sequences for interpolation in the subspace

Connects zero sets of Blaschke products with interpolation solvability

Advances understanding of function interpolation in Hardy space subspaces

Abstract

Given an interpolating Blaschke product $B$ with zeros $\{a_j\}$, we seek to characterize the sequences of values $\{w_j\}$ for which the interpolation problem $$f(a_j)=w_j\qquad (j=1,2,\dots)$$ can be solved with a function $f$ from the model subspace $H^1\cap B\overline{H^1_0}$ of the Hardy space $H^1$.