Traces of H\"ormander algebras on discrete sequences

TL;DR

This paper characterizes when a discrete sequence in the complex plane can be decomposed into interpolating sequences for H"ormander algebras, linking it to bounded divided differences of functions on the sequence.

Contribution

It provides a precise criterion for unions of interpolating sequences in H"ormander algebras based on divided differences, extending to Korenblum-type algebras in the unit disk.

Findings

Characterization of unions of interpolating sequences for $A_p$

Trace of $A_p$ matches functions with bounded divided differences

Extension of results to Korenblum-type algebras in the unit disk

Abstract

We show that a discrete sequence $\Lambda$ of the complex plane is the union of $n$ interpolating sequences for the H\"ormander algebras $A_p$ if and only if the trace of $A_p$ on $\Lambda$ coincides with the space of functions on $\Lambda$ for which the divided differences of order $n-1$ are uniformly bounded. The analogous result holds in the unit disk for Korenblum-type algebras.