Pólya frequency functions and sequences

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


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
$$ f\left( u \right) = {e^{ - \gamma {u^2}}} $$
where γ is a positive parameter, and the (truncated) exponential function
$$ \begin{array}{*{20}{c}} {f\left( u \right) = \left\{ {\begin{array}{*{20}{c}} {{e^{ - \lambda {\text{u}}}}}&{amp;u0} \\ 0&{amp;u < 0} \end{array}} \right.}&{amp;{\text{or}}}&{amp;f\left( v \right) = } \end{array}\left\{ {\begin{array}{*{20}{c}} {{e^{\lambda \nu }}}&{amp;\nu 0} \\ 0&{amp;\nu > 0} \end{array}} \right. $$
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.


Meromorphic Function Representation Formula Analyse Math Positive Kernel Monotone Likelihood Ratio 
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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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