Polyharmonic functions of infinite order on annular regions

TL;DR

This paper systematically studies polyharmonic functions of infinite order on annular regions, revealing their analytic extensions and providing constructive methods for such extensions based on Fourier-Laplace series.

Contribution

It introduces a systematic analysis of infinite order polyharmonic functions, including their analytic extension and constructive procedures, advancing understanding of their behavior on annular regions.

Findings

Fourier-Laplace coefficients extend analytically to the complex plane.

Constructive procedure for analytic extension to the harmonicity hull.

Methods rely on Taylor series analysis with differential operators.

Abstract

Polyharmonic functions f of infinite order and type {\tau} on annular regions are systematically studied. The first main result states that the Fourier-Laplace coefficients f_{k,l}(r) of a polyharmonic function f of infinite order and type 0 can be extended to analytic functions on the complex plane cut along the negative semiaxis. The second main result gives a constructive procedure via Fourier-Laplace series for the analytic extension of a polyharmonic function on annular region A(r_{0},r_{1}) of infinite order and type less than 1/2r_{1} to the kernel of the harmonicity hull of the annular region. The methods of proof depend on an extensive investigation of Taylor series with respect to linear differential operators with constant coefficients.