Hilbert spaces of entire functions with trivial zeros

TL;DR

This paper investigates Hilbert spaces of entire functions with trivial zeros and derives a method to compute the associated E-function for modified spaces where functions are divided by products of linear factors at specified zeros.

Contribution

It provides a formula to determine the E-function of a Hilbert space of entire functions after removing zeros at specified points, given the original E-function.

Findings

Derived a formula for the E-function of the modified space

Established the relationship between original and zero-removed spaces

Enhanced understanding of zero manipulation in Hilbert spaces of entire functions

Abstract

Let $H$ be a Hilbert space of entire functions. Let $H'$ be the space of the functions $f(z)/\prod_i(z-z_i)$ where $f$ belongs to $H$ and vanishes at $n$ given complex points $z_i$. We compute a suitable $E$ function for $H'$ when one is given for $H$.