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1. The result. Consider a point set C in the complex plane whose complement K is connected and regular (i.e. K possesses a Green's function with pole at infinity). Let d^ denote the transfinite diameter of C. Recall that d„ = 1/1 #'( oo ) | where <f>(z) maps K onto the exterior of the unit circle. Equivalently, d00 = Limn^oo[||2 n(2)||c] where lk()||c=max|g(2;)|, s £ C and Tn(z) is the Tchebycheff polynomial (or Faber polynomial) for C. Given ƒ(z) defined on C, let P*(z) be the best polynomial approximation tof(z) on C i.e., ||jf(*)-P»0s)||c is minimized for polynomials Pn(z) of degree n. The purpose of this note is to announce the

A characterization of entire functions in terms of degree of convergence