, Volume 19, Issue 4, pp 903–910 | Cite as

On some positive definite functions



We study the function \((1 - \Vert x\Vert )/ (1 - \Vert x\Vert ^r),\) and its reciprocal, on the Euclidean space \(\mathbb {R}^n,\) with respect to properties like being positive definite, conditionally positive definite, and infinitely divisible.


Positive definite Conditionally negative definite Infinitely divisible Operator monotone Completely monotone 

Mathematics Subject Classification

42A82 42B99 



The work of the first author is supported by a J. C. Bose National Fellowship, and of the second author by an SERB Women Excellence Award. The first author was a Fellow Professor at Sungkyunkwan University in the summer of 2014.


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Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.Indian Statistical InstituteNew DelhiIndia
  2. 2.Sungkyunkwan UniversitySuwonKorea

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