Interpolation by entire functions with growth conditions

TL;DR

This paper characterizes the trace of entire functions with specific growth conditions on discrete sequences using $L^2$ estimates for the $ar ext{ extbackslash}partial$ equation, advancing understanding of interpolation in complex analysis.

Contribution

It provides a new characterization of the trace of growth-restricted entire functions on discrete sets via $ar ext{ extbackslash}partial$ estimates, which is a novel approach.

Findings

Characterization of the trace of $A_p( ext{ extbackslash}C)$ on non-uniqueness sequences.

Application of $L^2$ estimates for the $ar ext{ extbackslash}partial$ equation in interpolation theory.

Insight into growth conditions for entire functions and their interpolation properties.

Abstract

Let $A_p(\C)$ be the space of entire functions such that $| f(z)|\le Ae^{Bp(z)}$ for some $A,B>0$ and let $V$ be a discrete sequence of complex numbers which is not a uniqueness set for $A_p(\C)$. We use $L^2$ estimates for the $\bar\partial$ equation to charaterize the trace of $A_p(\C)$ on $V$.