I. J. Schoenberg Selected Papers pp 321-330 | Cite as

# On Pólya frequency functions. II: Variation-diminishing integral operators of the convolution type

Chapter

## Abstract

A real matrix is variation-diminishing in the sense that if

*A*=∥*a*_{ ik }∥ (*i*=1,*m*;*k*= 1,...,*n*) is said to be*totally positive*if all its minors, of any order, are non-negative. In 1930^{2}) the author showed that if*A*is totally positive, then the linear transformation$$
{y_i} = \sum\limits_{k = 1}^n {{a_{ik}}{x_k}} \quad \left( {i = 1, \ldots ,m} \right)
$$

(1)

*v*(*x*_{ k }) denotes the number of variations of sign in the sequence*x*_{ k }and*v*(*y*_{ i }) the corresponding number in the sequence*y*_{ i }, then we always have the inequality*v*(*y*_{ i }) ≦*v*(*x*_{ k }). In the same paper of 1930 the author showed that (1) is certainly variationdiminishing if the matrix*A*does not possess two minors of equal orders and of opposite signs; also the converse holds to a certain extent: If (1) is variation-diminishing, then*A*cannot have two minors of equal orders and of opposite signs,*provided the rank of A is*=*n.*The necessary and sufficient conditions in order that (1) be variation-diminishing were found in 1933 by Th. Motzkin^{3}). Since they will be used in this paper we state them here as follows: Let*r*be the rank of*A*then*A*should not have two minors of equal orders and of opposite signs if their common order is <*r*, while if their common order is =*r*then again they should never be of opposite signs if they belong to the same combination of*r*columns of*A*.## Keywords

Opposite Sign Frequency Function Real Zero Direct Part Convolution Type
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## References

- 1).A résumé of the results of this paper has appeared under the same title in the
*Proceedings of the National Academy of Sciences*,**34**(1948), pp. 164-169.Google Scholar - 1).I. J. Schoenberg, Über variationsvermindernde lineare Transformationen,
*Math. Zeitschrift*,**32**(1930), pp. 321–328.CrossRefGoogle Scholar - 3).Th. Motzkin,
*Beiträge zur Theorie der linearen Ungleichungen*, Doctoral dissertation, Basel, 1933 (Jerusalem, 1936), 69 pp., especially Chap. IV.Google Scholar - 4).I. J. Schoenberg, On totally positive functions, Laplace integrals and entire functions of the Laguerre-Pólya-Schur type,
*Proceedings of the National Academy of Sciences*,**33**(1947), pp. 11–17. A detailed paper will appear under the title “On Pólya frequency functions. I: Totally positive functions and their Laplace transforms” probably in the*Transactions of the American Mathematical Society*.CrossRefGoogle Scholar - 6).G. Pólya, Algebraische Untersuchungen über ganze Funktionen vom Geschlechte Null und Eins,
*Journal für die reine und angewandte Math.*,**145**(1915), pp. 224–249, especially p. 231.Google Scholar

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© Springer Science+Business Media New York 1988