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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Function spaces for conditionally positive definite operator-valued kernels


Authors: Georg Berschneider, Wolfgang zu Castell and Stefan J. Schrödl
Journal: Math. Comp. 81 (2012), 1551-1569
MSC (2010): Primary 46C20, 41A63, 47A56; Secondary 41A05, 62M30, 62H30
Published electronically: October 18, 2011
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Abstract: The correspondence between reproducing kernel Hilbert spaces and positive definite kernels is well understood since Aronszajn's work of the 1940s. The analog relation is less clear for conditionally positive definite kernels. The latter are widely used in approximation methods for scattered data, geostatistics, and machine learning. We consider this relation and provide two ways to construct a reproducing kernel Pontryagin space for operator-valued kernels.


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Additional Information

Georg Berschneider
Affiliation: Institute for Mathematical Stochastics, Technische Universität Dresden, 01062 Dresden, Germany
Email: georg.berschneider@tu-dresden.de

Wolfgang zu Castell
Affiliation: Department of Scientific Computing, Helmholtz Zentrum München, German Research Center for Environmental Health, Ingolstaedter Landstrasse 1, 85764 Neuherberg, Germany
Email: castell@helmholtz-muenchen.de

Stefan J. Schrödl
Affiliation: Department of Scientific Computing, Helmholtz Zentrum München, German Research Center for Environmental Health, Ingolstaedter Landstrasse 1, 85764 Neuherberg, Germany
Email: stefan.schroedl@gmx.net

DOI: http://dx.doi.org/10.1090/S0025-5718-2011-02552-4
PII: S 0025-5718(2011)02552-4
Received by editor(s): April 8, 2010
Received by editor(s) in revised form: March 8, 2011
Published electronically: October 18, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.