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- W2061881543 abstract "An 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<1/q, then there exist z-harmonic functions kz such that μ1nf(v)=∫Iznkz(v)dz, 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 with C⩾1/q, which necessitates replacing kz(v)dz with an analytic distribution νv satisfying the z-harmonicity condition μ1ν=zν. We show that to each function on the tree satisfying the above exponential growth condition there corresponds an eigendistribution on an elliptical region containing I as the interval between its foci." @default.
- W2061881543 created "2016-06-24" @default.
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- W2061881543 date "2002-11-01" @default.
- W2061881543 modified "2023-09-26" @default.
- W2061881543 title "Analytic eigendistributions and applications to homogeneous trees" @default.
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- W2061881543 doi "https://doi.org/10.1016/s0196-8858(02)00031-3" @default.
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