Near-Circularity in Capacity and Maximally Convergent Polynomials

Abstract

AbstractIf 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 $$\left\{ p_n\right\} _{n\in \mathbb {N}}$$ 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 $$f-p_n$$ f - p n is asymptotically constant in capacity on level lines of the Green’s function $$g_\Omega (z,\infty )$$ g Ω ( z , ∞ ) of the complement $$\Omega $$ Ω of E in $$\overline{\mathbb {C}}$$ 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.