Vol. 67, No. 1, 1976

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Total positivity of certain reproducing kernels

Jacob Burbea

Vol. 67 (1976), No. 1, 101–130

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.

Mathematical Subject Classification 2000
Primary: 46E20
Secondary: 30A98
Received: 4 December 1975
Published: 1 November 1976
Jacob Burbea