Near-Circularity in Capacity and Maximally Convergent Polynomials

If f is a power series with radius R of convergence, $$R > 1$$ R > 1 , it is well-known that the method of Carathéodory–Fejér constructs polynomial approximations of f on the closed unit disk which show the typical phenomenon of near-circularity on the unit circle. Let E be compact and connected and let f be holomorphic on E . If \ pₙ\ n∈ N p n n ∈ N is a sequence of polynomials converging maximally to f on E , it is shown that the modulus of the error functions f-pₙ f - p n is asymptotically constant in capacity on level lines of the Green’s function g_Ω (z,∞ ) g Ω ( z , ∞ ) of the complement Ω Ω of E in C̄ C ¯ with pole at infinity, thereby reflecting a type of near-circularity, but without gaining knowledge of the winding numbers of the error curves with respect to the point 0.