# Strictly positive definite kernels on a product of circles

Strictly positive definite kernels on a product of circles We supply a Fourier characterization for the real, continuous, isotropic and strictly positive definite kernels on a product of circles. In other words, if $$S^1$$ S 1 is the unit circle in $$\mathbb {R}^2$$ R 2 , $$\cdot$$ · is the usual inner product of $$\mathbb {R}^2$$ R 2 and f is a real continuous function on $$[-1,1]^2$$ [ - 1 , 1 ] 2 , we determine necessary and sufficient conditions in order that $$f(x\cdot y,z \cdot w)$$ f ( x · y , z · w ) be a strictly positive definite kernel on $$S^1 \times S^1$$ S 1 × S 1 . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# Strictly positive definite kernels on a product of circles

Positivity, Volume 21 (1) – May 12, 2016
14 pages

/lp/springer_journal/strictly-positive-definite-kernels-on-a-product-of-circles-P50UyX5YwB
Publisher
Springer Journals
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/s11117-016-0425-1
Publisher site
See Article on Publisher Site

### Abstract

We supply a Fourier characterization for the real, continuous, isotropic and strictly positive definite kernels on a product of circles. In other words, if $$S^1$$ S 1 is the unit circle in $$\mathbb {R}^2$$ R 2 , $$\cdot$$ · is the usual inner product of $$\mathbb {R}^2$$ R 2 and f is a real continuous function on $$[-1,1]^2$$ [ - 1 , 1 ] 2 , we determine necessary and sufficient conditions in order that $$f(x\cdot y,z \cdot w)$$ f ( x · y , z · w ) be a strictly positive definite kernel on $$S^1 \times S^1$$ S 1 × S 1 .

### Journal

PositivitySpringer Journals

Published: May 12, 2016

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