Total positivity of certain reproducing kernels

Abstract

In this paper we study the total positivity of variouskernels, especially reproducing kernels of Hubert spaces ofanalytic functions. We do so by employing a familiar deviceknown as the “ composition formula of Pólya and Szegö.”Using this formula we are able to give a short proof of thevariation diminishing property of a generalized analogue ofthe la Vallée Poussin means. This generalizes earlier workof Pólya and Schoenberg and recent work of Horton. Ourmethod is also based on the isometrical image of the reproducingkernel called the generating function. The reproducingkernel is then expressed as a composition of twogenerating functions so that the problem is reduced toinvestigating the total positivity of the generating function.This methods extends earlier work and yields many new reproducingkernels which are total positive. © 1976, University of California, Berkeley. All Rights Reserved.