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Robin functions and extremal functions

T. BloomN. LevenbergS. Ma'u — 2003

Annales Polonici Mathematici

Given a compact set K N , for each positive integer n, let V ( n ) ( z ) = V K ( n ) ( z ) := sup 1 / ( d e g p ) V p ( K ) ( p ( z ) ) : p holomorphic polynomial, 1 ≤ deg p ≤ n. These “extremal-like” functions V K ( n ) are essentially one-variable in nature and always increase to the “true” several-variable (Siciak) extremal function, V K ( z ) := max[0, sup1/(deg p) log|p(z)|: p holomorphic polynomial, | | p | | K 1 ]. Our main result is that if K is regular, then all of the functions V K ( n ) are continuous; and their associated Robin functions ϱ V K ( n ) ( z ) : = l i m s u p | λ | [ V K ( n ) ( λ z ) - l o g ( | λ | ) ] increase to ϱ K : = ϱ V K for all z outside a pluripolar set....

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