Analytic eigendistributions and applications to homogeneous trees

AbstractAn analytic distribution on K⊆C is an element, ν, of the dual of the space of analytic functions on K. In particular, ν defines a linear functional on the polynomial ring C[z]. In this work, we study the converse problem: given a linear functional on C[z], try to find a minimal set K such that ν extends to an analytic distribution on K. This study was motivated by the desire to generalize a result that allows the representation of functions on a homogeneous tree as integrals of z-harmonic functions oven a certain interval. A function f on a homogeneous tree T of degree q+1 is said to be z-harmonic, if μ1f=zf, where μ1 is the nearest neighbor averaging operator. It was proved in [Cohen, Colonna, Adv. Appl. Math. 20 (1998) 253–274] that if |f(v)|⩽MC|v| for constants M>0 and 0<C