On Pólya Frequency Functions. III. The Positivity of Translation Determinants with an Application to the Interpolation Problem by Spline Curves

Abstract

1. A frequency function Δ(x), i. e., a non-negative measurable function satisfying the inequalities

$$ 0 < \int_{ - \infty }^\infty {\Lambda \left( x \right)dx < } \infty , $$

is called a Pólya frequency function provided(²) it satisfies the following condition: For every two sets of increasing numbers

$$ \begin{array}{*{20}{c}} {{x_1} < {x_2} < \cdots < {x_n},}&{amp;{y_1} < {y_2} < \cdots < {y_n},}&{amp;n = 1,{\mkern 1mu} 2,{\mkern 1mu} \cdots ,} \end{array} $$

(1)

we have the inequality

$$ D \equiv \det {\left\| {\Lambda \left( {{x_i} - {y_i}} \right)} \right\|_{1,n}}0. $$

(2)

.

This work was performed on a National Bureau of Standards contract with the University of California at Los Angeles and was sponsored in part by the Office of Scientific Research, USAF. Miss Whitney’s contribution to this paper was accepted by the Graduate School of the University of Pennsylvania in partial fulfilment of the requirement for the Ph.D. degree. For a brief summary see [7] in the list of references at the end of this paper.