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Analytic eigendistributions and applications to homogeneous trees

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An analytic distribution onK⊆ℂ is an element,, of the dual of the space of analytic functions on K. In particular, defines a linear functional on the polynomial ring ℂ[z]. In this work, we study the converse problem: given a linear functional onℂ[z], try to find a minimal setK such that extends to an analytic distribution onK. 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 functionf on a homogeneous treeT of degreeq+1 is said to bez-harmonic, if1f=zf, where1 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 and0z such that\mu _{\scriptscriptstyle{\mathrm{1}}}^{\scriptscriptstyle{n}}f\left( v\right) ={\underset{\scriptstyle{I}}{\int }}z^{\scriptscriptstyle{{n}}}k_{\scriptscriptstyle{{z}}}\left( v\right) \hspace{0.2em}\mathrm{{d}}z, where I is the interval with endpoints┬▒2q/(q+1). In the present paper, we study the case when the above exponential growth condition holds withC1/q, which necessitates replacing kz(v)dz with an analytic distributionv satisfying thez-harmonicity condition1=z. We show that to each function on the tree satisfying the above exponential growth condition there corresponds an eigendistribution on an elliptical region containingI as the interval between its foci.

┬ę 2002 Elsevier Science (USA)

Document Type: Research Article

DOI: http://dx.doi.org/10.1016/S0196-8858(02)00031-3

Affiliations: 1: University of Maryland, College Park, MD, USA 2: George Mason University, Fairfax, VA, USA

Publication date: November 1, 2002

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