Near-Circularity in Capacity and Maximally Convergent Polynomials

Abstract

If f is a power series with radius R of convergence, \(R > 1\), 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 \(\left\{ p_n\right\} _{n\in \mathbb {N}}\) is a sequence of polynomials converging maximally to f on E, it is shown that the modulus of the error functions \(f-p_n\) is asymptotically constant in capacity on level lines of the Green’s function \(g_\Omega (z,\infty )\) of the complement \(\Omega \) of E in \(\overline{\mathbb {C}}\) 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.