Polyharmonic functions of infinite order on annular regions

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.