This article is cited in 3 papers

A theorem on order relations generated by totally positive kernels

M. L. Gerver^a, E. A. Kudryavtseva<sup>b

a</sup> International Institute of Earthquake Prediction Theory and Mathematical Geophysics RAS
^b M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: This paper is a result of reflections on a theorem on 'sums of hyperbolae'$^1$ needed in mathematical geophysics. This theorem and several unexpected corollaries of it do not seem very plausible at first sight. The first proof of the theorem did not dispel this impression: the reasons why it was true remained obscure. The explanation was found in the properties of the Cauchy kernel $C(s,x)=1/(s+x)$. In this paper the original theorem on 'hyperbolae' is established as a particular case of a general result holding for a certain class $\mathbb G$ of totally positive kernels which contains $C(s,x)$.

UDC: 517+550.34

MSC: Primary 26C15, 26D20; Secondary 41A50, 73D50, 73N10

Received: 29.12.1994

 English version: Sbornik: Mathematics, 1995, 186:9, 1241–1269

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