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

Abstract

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

$$ {y_i} = \sum\limits_{k = 1}^n {{a_{ik}}{x_k}} \quad \left( {i = 1, \ldots ,m} \right) $$

(1)

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.