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
Part of the Contemporary Mathematicians book series (CM)


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
$$ \begin{array}{*{20}{c}} {{x_1} < {x_2} < \cdots < {x_n},}&{amp;{y_1} < {y_2} < \cdots < {y_n},}&{amp;n = 1,{\mkern 1mu} 2,{\mkern 1mu} \cdots ,} \end{array} $$
we have the inequality
$$ D \equiv \det {\left\| {\Lambda \left( {{x_i} - {y_i}} \right)} \right\|_{1,n}}0. $$


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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    F. Gantmakher and M. Krein, Oscillatory matrices and kernels and small vibrations of mechanical systems (in Russian), 2d ed., Moscow, 1950.Google Scholar
  2. 2.
    H. Hahn, Über das Interpolationsproblem, Math. Zeit. vol. 1 (1918) pp. 115–142.CrossRefGoogle Scholar
  3. 3.
    I. I. Hirschman and D. V. Widder, The inversion of a general class of convolution transforms, Trans. Amer. Math. Soc. vol. 66 (1949) pp. 135–201.CrossRefGoogle Scholar
  4. 4.
    M. Krein and G. Finkelstein, Sur les fonctions de Green complètement non-négatives des opérateurs différentiels ordinaires, C. R. (Doklady) Acad. Sci. URSS. vol. 24 (1939) pp. 220–223.Google Scholar
  5. 5.
    I. J. Schoenberg, Contributions to the problem of approximation of equidistant data by analytic functions, Parts A and B, Quarterly of Applied Mathematics vol. 4 (1946) pp. 45–99, 112-141.Google Scholar
  6. 6.
    —, On Pólya frequency functions I. The totally positive functions and their Laplace transforms, Journal d’Analyse Mathématique vol. 1 (1951) pp. 331–374.CrossRefGoogle Scholar
  7. 7.
    I. J. Schoenberg and Anne Whitney, Sur la positivité des déterminants de translations des fonctions de fréquence de Pólya avec une application au problème d’interpolation par les fonctions “spline,” C. R. Acad. Sci. Paris vol. 228 (1949) pp. 1996–1998.Google Scholar

Copyright information

© Springer Science+Business Media New York 1988

Authors and Affiliations

  • I. J. Schoenberg
    • 1
  • Anne Whitney
    • 1
  1. 1.University of PennsylvaniaPhiladelphiaUSA

Personalised recommendations