Planar Lebesgue Measure of Exceptional Set in Approximation of Subharmonic Functions

TL;DR

This paper investigates how well subharmonic functions can be approximated by entire functions, providing lower bounds on the measure of the set where the approximation fails, especially for functions of finite order.

Contribution

It offers a new estimate for the Lebesgue measure of the exceptional set in the approximation of subharmonic functions by entire functions of finite order.

Findings

Lower bounds on the Lebesgue measure of exceptional sets

Approximation results for subharmonic functions of finite order

Quantitative measure estimates in approximation theory

Abstract

We consider the pointwise approximation of a subharmonic function by the logarithm of the modulus of an entire function up to a bounded quantity. In the case of finite order an estimate from below of the planar Lebesgue measure of an exceptional set in such approximation is obtained.