Inner functions and zero sets for $\ell^{p}_{A}$

TL;DR

This paper characterizes zero sets of functions in $\,ell^{p}_A$ spaces using a new concept of inner functions, revealing zero sets beyond classical Blaschke sequences, especially for p > 2.

Contribution

It introduces a novel inner function concept for $\,ell^{p}_A$ spaces and provides zero set criteria that extend classical results.

Findings

Zero sets for $\,ell^{p}_A$ are characterized using inner functions.

For p > 2, there exist zero sets not covered by classical Blaschke sequences.

New families of zero sets are constructed beyond traditional theorems.

Abstract

In this paper we characterize the zero sets of functions from $\ell^{p}_{A}$ (the analytic functions on the open unit disk $D$ whose Taylor coefficients form an $\ell^p$ sequence) by developing a concept of an "inner function" modeled by Beurling's discussion of the Hilbert space $\ell^{2}_{A}$ (the classical Hardy space). The zero set criterion is used to construct families of zero sets which are not covered by classical results. In particular, it is proved that when $p > 2$, there are zero sets for $\ell^{p}_{A}$ which are not Blaschke sequences.