On Pólya frequency functions. II: Variation-diminishing integral operators of the convolution type
I. J. Schoenberg · Birkhäuser Boston eBooks · 1988
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 1 y i = ∑ k=1 n a ik x k ( i=1,…,m ) $$ {y_i} = \sum\limits_{k = 1}^n {{a_{ik}}{x_k}} \quad \left( {i = 1, \ldots ,m} \right) $$ 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.