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Pólya frequency functions and sequences

  • Samuel Karlin
Part of the Contemporary Mathematicians book series (CM)

Abstract

The highly important totally positive kernels of the form K(x, y) = f(x−y) where x, y traverse the real line (or the integers), Schoenberg called Pólya frequency (PF) functions (sequences). The two prime examples of PF functions are the normal function
$$ f\left( u \right) = {e^{ - \gamma {u^2}}} $$
(1)
where γ is a positive parameter, and the (truncated) exponential function
$$ \begin{array}{*{20}{c}} {f\left( u \right) = \left\{ {\begin{array}{*{20}{c}} {{e^{ - \lambda {\text{u}}}}}&{amp;u0} \\ 0&{amp;u < 0} \end{array}} \right.}&{amp;{\text{or}}}&{amp;f\left( v \right) = } \end{array}\left\{ {\begin{array}{*{20}{c}} {{e^{\lambda \nu }}}&{amp;\nu 0} \\ 0&{amp;\nu > 0} \end{array}} \right. $$
(2)
where λ is a free positive parameter. In a remarkable series of papers, Schoenberg (see his review paper (1953) [48*]) set the basis of the theory of Pólya frequency functions, and established the fundamental representation theorems. Earlier works of Pólya, Laguerre and Schur aided these developments; their concern was the approximation of functions by polynomials with only real zeros.

Keywords

Meromorphic Function Representation Formula Analyse Math Positive Kernel Monotone Likelihood Ratio 
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.

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Copyright information

© Springer Science+Business Media New York 1988

Authors and Affiliations

  • Samuel Karlin
    • 1
  1. 1.Stanford UniversityUSA

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