Total Positivity and Variation Diminishing Transformations

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


I. J. Schoenberg has made outstanding mathematical contributions in several areas of analysis. In particular, he fathered the subject of total positivity and continuously nurtured it with highly original and versatile research for more than five decades. I will review in this essay the concept of total positivity and the associated variation diminishing property emphasizing its wide scope in applications and its symbiotic mathematical relationships. I will also comment on recent developments (e. g., multivariate generalizations, connections to combinatorics) of this topic. In a companion essay I briefly discuss the important class of translation total positive kernels, the Pólya frequency functions.


Total Positivity Positive Kernel Compound Probability Correlation Inequality Wishart Matrice 
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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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