Afterword

Summary

It is very difficult, if not well nigh impossible, to give an exact history of the development of any set of ideas. Nonetheless, there are four persons whose contributions “stand out” when considering the history of total positivity. They are I. J. Schoenberg, M. G. Krein, F. R. Gantmacher, and S. Karlin. Of course they did not work in a vacuum and numerous influences are very evident in their research.

It was Schoenberg who initiated the study of the variation diminishing properties of totally positive matrices in 1930 in Schoenberg [1930], and the study of Pólya frequency functions in the late 1940s and early 1950s. Independently, and unaware of Schoenberg's work, Krein was developing the theory of total positivity as it related to ordinary differential equations whose Green's functions are totally positive. Furthermore, in the mid- 1930s Krein, together with Gantmacher, proved the spectral properties of totally positive kernels and matrices, and many other properties (see Gantmacher, Krein [1935], Gantmacher [1936], Gantmacher, Krein [1937], and their influential Gantmacher, Krein [1941], which was later reissued as Gantmacher, Krein [1950], and its translations in German in 1960 and in English in 1961 and 2002). These topics are the foundations upon which has been constructed the theory of total positivity. Karlin's role was somewhat different. His books Karlin, Studden [1966] and Karlin [1968], the latter titled Total Positivity. Volume 1 (but there is no Volume 2), presented many new results and ideas and also synthesized and popularized many of these ideas.