Remote Access Mathematics of Computation
Green Open Access

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
MathSciNet review: 2904590
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • 1. D. Alpay, A. Dijksma, J. Rovnyak, and H. de Snoo, Schur functions, operator colligations, and reproducing kernel Pontryagin spaces, Operator Theory: Advances and Applications, vol. 96, Birkhäuser, Basel, 1997. MR 1465432 (2000a:47024)
  • 2. L. Amodei and M. N. Benbourhim, A vector spline approximation, J. Approx. Theory 67 (1991), no. 1, 51-79. MR 1127820 (92i:41012)
  • 3. G. Berschneider and W. zu Castell, Conditionally positive definite kernels and Pontryagin spaces, Approximation Theory XII. Proceedings of the 12th International Conference, San Antonio, TX, USA, March 4-8, 2007 (M. Neamtu and L. L. Schumaker, eds.), Modern Methods in Mathematics, Nashboro Press, Brentwood, TN, 2008, pp. 27-37. MR 2537117 (2010j:41001)
  • 4. J. Bognár, Indefinite inner product spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 78, Springer, Berlin-Heidelberg-New York, 1974. MR 0467261 (57:7125)
  • 5. W. Cheney and W. Light, A course in approximation theory, Brooks/Cole, Pacific Grove, CA, 1999.
  • 6. J.-P. Chilès and P. Delfiner, Geostatistics. Modeling spatial uncertainty, Wiley Series in Probability and Statistics, John Wiley and Sons, New York, NY, 1999. MR 1679557 (2000f:86010)
  • 7. F. Dodu and C. Rabut, Vectorial interpolation using radial-basis-like functions, Comput. Math. Appl. 43 (2002), no. 3-5, 393-411. MR 1883575 (2003a:65010)
  • 8. T. Evgeniou, C. A. Micchelli, and M. Pontil, Learning multiple tasks with kernel methods,
    J. Mach. Learn. Res. 6 (2005), 615-637. MR 2249833
  • 9. E. J. Fuselier, Improved stability estimates and a characterization of the native space for matrix-valued RBFs, Adv. Comput. Math. 29 (2008), no. 3, 311-313. MR 2438347 (2009k:41003)
  • 10. D. Handscomb, Local recovery of a solenoidal vector field by an extension of the thin-plate spline technique, Numer. Algorithms 5 (1993), no. 1-4, 121-129. MR 1258589
  • 11. W. Light and H. Wayne, Spaces of distributions, interpolation by translates of a basis function and error estimates., Numer. Math. 81 (1999), no. 3, 415-450. MR 1668091 (99m:65021)
  • 12. G. B. Pedrick, Theory of reproducing kernel Hilbert spaces of vector-valued functions, Studies in Eigenvalue Problems 19, University of Kansas, Department of Mathematics, 1957.
  • 13. Z. Sasvári, Positive definite and definitizable functions, Mathematical Topics, vol. 2, Akademie Verlag, Berlin, 1994. MR 1270018 (95c:43005)
  • 14. R. Schaback, A unified theory of radial basis functions. Native Hilbert spaces for radial basis functions II, J. Comput. Appl. Math. 121 (2000), no. 1-2, 165-177. MR 1752527 (2002h:41037)
  • 15. B. Schölkopf and A. J. Smola, Learning with kernels - support vector machines, regularization, optimization, and beyond, MIT Press, Cambridge, MA, 2002.
  • 16. L. Schwartz, Sous-espaces hilbertiens d'espaces vectoriels topologiques et noyaux associés. (Noyaux reproduisants), J. Anal. Math. 13 (1964), 115-256 (French). MR 0179587 (31:3835)
  • 17. P. Sorjonen, Pontrjaginräume mit einem reproduzierenden Kern, Ann. Acad. Sci. Fenn., Ser. I. A. Math. 594 (1975) (German). MR 0405079 (53:8875)
  • 18. H. Wendland, Scattered data approximation, Cambridge Monographs on Applied and Computational Mathematics, vol. 17, Cambridge University Press, Cambridge, 2005. MR 2131724 (2006i:41002)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 46C20, 41A63, 47A56, 41A05, 62M30, 62H30

Retrieve articles in all journals with MSC (2010): 46C20, 41A63, 47A56, 41A05, 62M30, 62H30

Additional Information

Georg Berschneider
Affiliation: Institute for Mathematical Stochastics, Technische Universität Dresden, 01062 Dresden, Germany

Wolfgang zu Castell
Affiliation: Department of Scientific Computing, Helmholtz Zentrum München, German Research Center for Environmental Health, Ingolstaedter Landstrasse 1, 85764 Neuherberg, Germany

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

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.

American Mathematical Society