A Refinement of Carlson's Theorem

TL;DR

This paper refines Carlson's theorem by providing more precise growth estimates for analytic functions along the imaginary axis, incorporating spectral measure conditions.

Contribution

It introduces necessary and sufficient spectral measure conditions for the growth of functions, extending Carlson's theorem.

Findings

Refined growth estimates for analytic functions along the imaginary axis.

Spectral measure conditions characterize function growth.

Enhanced understanding of zero distribution and spectral properties.

Abstract

Carlson's theorem estimates the growth of an analytic function along the imaginary axis, provided that the function is zero at non-negative integers. We refine this theorem and describe not only the function's growth but also necessary and sufficient conditions in terms of its spectral measure.