Analytic continuation by reproducing kernel methods combined with Padé approximations

Abstract

The analytic continuation of power series is an old problem attacked by various methods, a notable one being the Padé approximant. Although quite powerful in some cases, the Padé approximant suffers sometimes from being a non-linear transformation. The linearity is useful whenever the coefficients of the Taylor developments are themselves functions of another complex variable. There are well-known linear transformations that improve convergence and their connection with some conformal mapping was discovered long ago, although not always appreciated. The present paper endeavours to extend the applicability of such methods by means of reproducing kernels. A general and flexible analytic continuation method — which does not have the drawback of limiting processes — is outlined, shown to encompass other existing procedures and to be potentially a strong competitor to the Padé approximant. The dynamic polarizability of hydrogen is shown as a numerical example.