Dolomites Research Notes on Approximation

We define an analog of the Leja-Siciak-Zaharjuta subharmonic extremal function for a proper subset E of the Berkovich projective line P1 over a field with a non-archimedean absolute value, relative to a point ζ ̸∈ E. When E is a compact set with positive capacity we prove that the upper semicontinuous regularization of this extremal function equals the Green function of E relative to ζ. As a separate result, we prove the Brelot-Cartan principle, under the additional assumption that the Berkovich topology is second countable