Total positivity and reproducing kernels

Abstract

In this paper we investigate the relationship between total positivity and reproducing kernels. We extend the notion of total positivity to domains in the complex plane. In doing so, we also give a geometrical interpretation to certain Wronskians of reproducing kernels. These geometrical quantities are connected to Gaussian curvatures of Kahler metrics induced by these kernels. For simply-connected domains these curvatures are negative constants, thereby showing that the kernels are totally positive and moreover yielding an efficient method for computing the relevant determinants. In general, the reproducing kernels of multiplyconnected domains are not totally positive. The motivation for this work stems from the work of Karlin [7] which deals with quadrature formulas. Let H be a Hubert space of functions analytic in a plane domain D and possessing a reproducing kernel K(z, t), z,teD. Let LeH*, where iϊ* is the dual of H. A subset & of H* is specified and a member Q e & is called a quadrature formula. To each Q e & is associated a remainder functional RQ = L — Q. An optimal quadrature formula, if it exists, is any member Q* e ^ satisfying