In this paper we study the total
positivity of various kernels, especially reproducing kernels of Hilbert spaces of
analytic functions. We do so by employing a familiar device known as the
“composition formula of Pólya and Szegő.” Using this formula we are able to give a
short proof of the variation diminishing property of a generalized analogue of the la
Vallée Poussin means. This generalizes earlier work of Pólya and Schoenberg and
recent work of Horton. Our method is also based on the isometrical image of the
reproducing kernel called the generating function. The reproducing kernel is then
expressed as a composition of two generating functions so that the problem is
reduced to investigating the total positivity of the generating function. This methods
extends earlier work and yields many new reproducing kernels which are total
positive.
