Abstract. In this paper, we introduce and study polyharmonic functions on trees. We prove that the discrete version of the classical Riquier problem can be solved on trees. Next, we show that the discrete version of a characterization of harmonic functions due to Globevnik and Rudin holds for biharmonic functions on trees. Furthermore, on a homogeneous tree we characterize the polyharmonic functions in terms of integrals with respect to finitely-additive measures (distributions) of certain functions depending on the Poisson kernel. We define polymartingales on a homogeneous tree and show that the discrete version of a characterization of polyharmonic functions due to Almansi holds for polymartingales. We then show that on homogeneous trees there are 1-1 correspondences among the space of nth-order polyharmonic functions, the space of nth-order polymartingales, and the space of n-tuples of distributions. Finally, we study the boundary behavior of polyharmonic functions on homogeneous trees whose associated distributions satisfy various growth conditions.

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Fatou-Naïm-Doob limit theorems in the axiomatic system of Brelot Annales de l'institut Fourier, tome 16, n o 2 (1966), p. 455-467 © Annales de l'institut Fourier, 1966, tous droits réservés. L'accès aux archives de la revue « Annales de l'institut Fourier » (http://annalif.ujf-grenoble.fr/) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/

Let Ω and Ω′ be two locally compact, connected Hausdorff spaces having countable bases. On each of the spaces is defined a system of harmonic functions satisfying the axioms of M. Brelot [2]. The following is the description of such a system. To each open set of Ω is assigned a vector space of finite continuous functions, called the harmonic functions, on this set.

Let/be a function on a product space XxY with values in a separable metrizable space such that it is measurable in one variable and continuous in the other. The joint measurability of such a function is proved under certain conditions on A" and T.