The Newman--Shapiro problem

TL;DR

This paper addresses the longstanding Newman--Shapiro problem in weighted approximation within the Fock space, providing a counterexample and exploring conditions for positive approximation results.

Contribution

It offers a negative answer to the Newman--Shapiro problem and presents new positive approximation results under specific restrictions.

Findings

Counterexample disproves the Newman--Shapiro conjecture.

Certain restrictions enable positive approximation results.

Advances understanding of weighted approximation in Fock space.

Abstract

We give a negative answer to the Newman--Shapiro problem on weighted approximation for entire functions formulated in 1966 and motivated by the theory of operators on the Fock space. There exists a function in the Fock space such that its exponential multiples do not approximate some entire multiples in the space. Furthermore, we establish several positive results under different restrictions on the function in question.