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Positivity

, Volume 20, Issue 1, pp 81–98 | Cite as

On determination of positive-definiteness for an anisotropic operator

  • Charles E. BakerEmail author
Article
  • 80 Downloads

Abstract

We study the positive-definiteness of a family of \(L^2(\mathbb {R})\) integral operators with kernel \(K_{t, a} (x, y) = \pi ^{-1} (1 + (x - y)^2+ a(x^2 + y^2)^t)^{-1}\), for \(t > 0\) and \(a > 0\). For \(0 < t \le 1\) and \(a > 0\), the known theory of positive-definite kernels and conditionally negative-definite kernels confirms positive-definiteness. For \(t > 1\) and a sufficiently large, the integral operator is not positive-definite. For t not an integer, but with integer odd part, the integral operator is not positive-definite.

Keywords

Integral operators Positive-definite kernels Asymptotics 

Mathematics Subject Classification

Primary 42A82 Secondary 47B65 

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Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.Department of MathematicsThe Ohio State UniversityColumbusUSA

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