Two-point Pade approximation to Herglotz-Riesz transforms

Pade approximation is the rational generalization of Hermite interpolating polynomial. On its own merits, it has earned a relevant place in the theory of constructive approximation. In this article, we will develop an exhaustive analysis of two-point Pade approximations to Herglotz-Riesz transforms. We study the convergence problem when the poles are partially preassigned. In this analysis, the Stieltjes polynomials on the unit circle naturally arise. Finally, some illustrative numerical examples are discussed.