On Pólya Frequency Functions

I. the Totally Positive Functions and their Laplace Transforms
  • I. J. Schoenberg
Part of the Contemporary Mathematicians book series (CM)


We denote by T 1 the class of entire functions which are limits, uniform in every finite domain, of real polynomials with only real non-positive zeros. Likewise we denote by T 2 the wider class of entire functions obtained if in the previous definition we only require that the approximating polynomials be real and have only real zeros. From the classical work of Laguerre and Pólya(2) we know that ϕ(s) ∈ T 1 if and only if ϕ(s) admits a representation of the form
$$ \begin{array}{*{20}{c}} {\Phi \left( s \right) = C{e^{\gamma s}}{s^m}\prod\limits_{v = 1}^\infty {\left( {1 + {\delta _v}s} \right)} ,} \\ {\left( {C\;real,\;\lambda \geqq 0,\quad {\delta _v} \geqq 0,\quad \sum {{\delta _v} < \infty } } \right),} \end{array} $$
and also that the elements ψ(s) of the class T 2 are characterized by the representation
$$ \begin{array}{*{20}{c}} {\psi \left( s \right) = C{e^{ - \gamma }}^{{s^2} + \delta s}{s^m}\prod\limits_{v = 1}^\infty {\left( {1 + {\delta _v}s} \right){e^{ - \delta vs}},} } \\ {\left( {C\;real,\gamma \geqq 0,\;\delta ,{\delta _v}\;resl,\sum {{\delta _v}2 < \infty } } \right).} \end{array} $$


Entire Function Positive Function Imaginary Axis Frequency Function Real Zero 
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Copyright information

© Springer Science+Business Media New York 1988

Authors and Affiliations

  • I. J. Schoenberg
    • 1
  1. 1.PhiladelphiaUSA

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