Spectral Analysis on Homogeneous Trees

• Published in: Advances in applied mathematics 1998-02, Vol.20 (2), p.253-274
• Main Authors: Cohen, Joel M, Colonna, Flavia
• Format: Article
• Language: English
• Subjects: Exact sciences and technology; Functions of a complex variable; Mathematical analysis; Mathematics; Potential theory; Sciences and techniques of general use; Several complex variables and analytic spaces
• ISSN: 0196-8858; 1090-2074
• DOI: 10.1006/aama.1997.0570

For each complex numberz, we construct an operatorHzdefined on the space of all complex-valued functions on a homogeneous tree. This operator has the property that if a functionfis harmonic (i.e., the local averaging operator fixes the values off), thenHz(f) isz-harmonic (i.e., the local averaging operator multipliesHz(f) byz). Because the Laplacian is the local averaging operator minus the identity, az-harmonic function is an eigenfunction of the Laplacian relative to the eigenvaluez−1. We show that allz-harmonic functions are in the image ofHz, and we compareHzto another well-known operator which converts harmonic functions toz-harmonic functions. We then study the problem of representing a function as the integral ofz-harmonic functions with respect to a distribution. In particular, if a function grows no faster than exponentially, then the distribution is of the formf(z,−)dλ(z), wheref(z,−) is az-harmonic function and λ is the Lebesgue measure in C. If the base of the growth is sufficiently small, the distribution is supported over the interval [equation]