Valiron-Titchmarsh' and Related Theorems for Subharmonic Functions in $\mathbb{R}^n$ With Masses on a Half-Line

TL;DR

This paper extends the Valiron-Titchmarsh theorem to subharmonic functions in higher dimensions, establishing conditions under which their asymptotic behavior can be precisely characterized, especially when associated with masses on a half-line.

Contribution

It generalizes classical theorems from complex analysis to higher-dimensional subharmonic functions with specific mass distributions, providing new asymptotic analysis tools.

Findings

Extension of Valiron-Titchmarsh theorem to $ nn$ for subharmonic functions.

Identification of conditions for regular asymptotic behavior of $u$ and $N$.

Establishment of limits for $rac{ log u(r)}{N(r)}$ implying asymptotic regularity.

Abstract

The Valiron-Titchmarsh theorem on asymptotic behavior of entire functions with negative zeros is extended to subharmonic functions in $\mathbb{R}^n, n\geq 3$. We also show that the existence of the limit $\lim_{\rti} \frac{\log u(r)}{N(r)}$, where $N(r)$ is the averaged counting function of a subharmonic function $u$ with the associated masses on a ray implies the regular asymptotic behavior separately for $u$ and for $N$.