Vol. 55, No. 2, 1974

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Total positivity and reproducing kernels

Jacob Burbea

Vol. 55 (1974), No. 2, 343–359

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 Kähler 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 multiply-connected domains are not totally positive.

Mathematical Subject Classification 2000
Primary: 30A31
Secondary: 41A55, 41A65
Received: 24 May 1974
Published: 1 December 1974
Jacob Burbea