Divided Differences & Restriction Operator on Paley-Wiener Spaces $PW_{tau}^{p}$ for $N-$Carleson Sequences

TL;DR

This paper extends the characterization of restriction operators on Paley-Wiener spaces to N-Carleson sequences, providing necessary and sufficient conditions for these operators to be isomorphisms involving divided differences.

Contribution

It generalizes previous results by Lyubarskii and Seip to N-Carleson sequences, broadening the class of sequences for which the restriction operator's isomorphism properties are fully characterized.

Findings

Necessary and sufficient conditions for isomorphism on N-Carleson sequences.

Extension of Carleson condition to N-Carleson sequences.

Characterization involving divided differences.

Abstract

For a sequence of complex numbers $\Lambda$ we consider the restriction operator $R_{\Lambda}$ defined on Paley-Wiener spaces $PW_{\tau}^{p}$ ($1<p<\infty$). Lyubarskii and Seip gave necessary and sufficient conditions on $\Lambda$ for $R_{\Lambda}$ to be an isomorphism between $PW_{\tau}^{p}$ and a certain weighted $l^{p}$ space. The Carleson condition appears to be necessary. We extend their result to $N-$Carleson sequences (finite unions of $N$ disjoint Carleson sequences). More precisely, we give necessary and sufficient conditions for $R_{\Lambda}$ to be an isomorphism between $PW_{\tau}^{p}$ and an appropriate sequence space involving divided differences.