On determination of positive-definiteness for an anisotropic operator

On determination of positive-definiteness for an anisotropic operator We study the positive-definiteness of a family of $$L^2(\mathbb {R})$$ L 2 ( R ) integral operators with kernel $$K_{t, a} (x, y) = \pi ^{-1} (1 + (x - y)^2+ a(x^2 + y^2)^t)^{-1}$$ K t , a ( x , y ) = π - 1 ( 1 + ( x - y ) 2 + a ( x 2 + y 2 ) t ) - 1 , for $$t > 0$$ t > 0 and $$a > 0$$ a > 0 . For $$0 < t \le 1$$ 0 < t ≤ 1 and $$a > 0$$ a > 0 , the known theory of positive-definite kernels and conditionally negative-definite kernels confirms positive-definiteness. For $$t > 1$$ 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

On determination of positive-definiteness for an anisotropic operator

Positivity , Volume 20 (1) – May 30, 2015
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Publisher
Springer International Publishing
Copyright
Copyright © 2015 by Springer Basel
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-015-0342-8
Publisher site
See Article on Publisher Site

Abstract

We study the positive-definiteness of a family of $$L^2(\mathbb {R})$$ L 2 ( R ) integral operators with kernel $$K_{t, a} (x, y) = \pi ^{-1} (1 + (x - y)^2+ a(x^2 + y^2)^t)^{-1}$$ K t , a ( x , y ) = π - 1 ( 1 + ( x - y ) 2 + a ( x 2 + y 2 ) t ) - 1 , for $$t > 0$$ t > 0 and $$a > 0$$ a > 0 . For $$0 < t \le 1$$ 0 < t ≤ 1 and $$a > 0$$ a > 0 , the known theory of positive-definite kernels and conditionally negative-definite kernels confirms positive-definiteness. For $$t > 1$$ 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.

Journal

PositivitySpringer Journals

Published: May 30, 2015

References

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