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

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

## 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)
.

## Keywords

Interpolation Problem Interpolation Point Frequency Function Interpolation Formula Spline Curve
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

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