# 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

, Volume 20 (1) – May 30, 2015
18 pages

/lp/springer_journal/on-determination-of-positive-definiteness-for-an-anisotropic-operator-0ca9qPueag
Publisher
Springer International Publishing
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

## You’re reading a free preview. Subscribe to read the entire article.

### DeepDyve is your personal research library

It’s your single place to instantly
that matters to you.

over 12 million articles from more than
10,000 peer-reviewed journals.

All for just $49/month ### Explore the DeepDyve Library ### Unlimited reading Read as many articles as you need. Full articles with original layout, charts and figures. Read online, from anywhere. ### Stay up to date Keep up with your field with Personalized Recommendations and Follow Journals to get automatic updates. ### Organize your research It’s easy to organize your research with our built-in tools. ### Your journals are on DeepDyve Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more. All the latest content is available, no embargo periods. ### DeepDyve Freelancer ### DeepDyve Pro Price FREE$49/month

\$360/year
Save searches from Google Scholar, PubMed
Create lists to organize your research
Export lists, citations