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

  • I. J. Schoenberg
Part of the Contemporary Mathematicians book series (CM)


A real matrix 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 19302) the author showed that if A is totally positive, then the linear transformation
$$ {y_i} = \sum\limits_{k = 1}^n {{a_{ik}}{x_k}} \quad \left( {i = 1, \ldots ,m} \right) $$
is variation-diminishing in the sense that if 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.


Opposite Sign Frequency Function Real Zero Direct Part Convolution Type 
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  1. 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
  2. 1).
    I. J. Schoenberg, Über variationsvermindernde lineare Transformationen, Math. Zeitschrift, 32 (1930), pp. 321–328.CrossRefGoogle Scholar
  3. 3).
    Th. Motzkin, Beiträge zur Theorie der linearen Ungleichungen, Doctoral dissertation, Basel, 1933 (Jerusalem, 1936), 69 pp., especially Chap. IV.Google Scholar
  4. 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
  5. 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

Copyright information

© Springer Science+Business Media New York 1988

Authors and Affiliations

  • I. J. Schoenberg
    • 1
  1. 1.PhiladelphiaUSA

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