Opuscula Math. 41, no. 3 (2021), 283-300

Opuscula Mathematica

New characterizations of reproducing kernel Hilbert spaces and applications to metric geometry

Daniel Alpay
Palle E.T. Jorgensen

Abstract. We give two new global and algorithmic constructions of the reproducing kernel Hilbert space associated to a positive definite kernel. We further present a general positive definite kernel setting using bilinear forms, and we provide new examples. Our results cover the case of measurable positive definite kernels, and we give applications to both stochastic analysis and metric geometry and provide a number of examples.

Keywords: reproducing kernel, positive definite functions, approximation, algorithms, measures, stochastic processes.

Mathematics Subject Classification: 46E22, 43A35.

Full text (pdf)

  1. R.A. Aliev, C.A. Gadjieva, Approximation of hypersingular integral operators with Cauchy kernel, Numer. Funct. Anal. Optim. 37 (2016), no. 9, 1055-1065.
  2. D. Alpay, M. Porat, Generalized Fock spaces and the Stirling numbers, J. Math. Phys. 59 (2018), 063509.
  3. D. Alpay, P. Jorgensen, R. Seager, D. Volok, On discrete analytic functions: Products, rational functions and reproducing kernels, J. Appl. Math. Comput. 41 (2013), 393-426.
  4. N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc. 68 (1950), 337-404.
  5. A. Ben-Israel, T.N. Greville, Generalized Inverses, 2nd ed., CMS Books in Mathematics, Ouvrages de Mathématiques de la SMC, vol. 15, Springer-Verlag, New York, 2003.
  6. J. Bouvrie, B. Hamzi, Kernel methods for the approximation of nonlinear systems, SIAM J. Control Optim. 55 (2017), no. 4, 2460-2492.
  7. S. Boyd, L. Vandenberghe, Convex Optimization, Cambridge University Press, Cambridge, 2004.
  8. H. Brezis, Analyse fonctionnelle, Masson, Paris, 1987.
  9. T.E. Duncan, Some applications of fractional Brownian motion to linear systems, [in:] System theory: modeling, analysis and control (Cambridge, MA, 1999), volume 518 of Kluwer Internat. Ser. Engrg. Comput. Sci., pages 97-105. Kluwer Acad. Publ., Boston, MA, 2000.
  10. I.M. Guel'fand, G.E. Shilov, Les distributions. Tome 1, Collection Universitaire de Mathématiques, no. 8, Dunod, Paris, 1972, Nouveau tirage.
  11. R.A. Horn, C.R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, 1994, Corrected reprint of the 1991 original.
  12. T. Kato, Perturbation Theory for Linear Operators, lassics in Mathematics, Springer-Verlag, Berlin, 1995, Reprint of the 1980 edition.
  13. H. Meschkovski, Hilbertsche Räume mit Kernfunktion, Springer-Verlag, 1962.
  14. J. Neveu, Processus aléatoires gaussiens, Number 34 in Séminaires de mathématiques supérieures. Les presses de l’université de Montréal, 1968.
  15. P. Saint-Pierre, Approximation of the viability kernel, Appl. Math. Optim. 29 (1994), no. 2, 187-209.
  16. S. Saitoh, Theory of Reproducing Kernels and its Applications, vol. 189, Longman Scientific and Technical, 1988.
  17. A.J. Smola, B. Schölkopf, On a kernel-based method for pattern recognition, regression, approximation and operator inversion, Algorithmica 22 (1998), no. 1-2, 211-231.
  18. E.M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, vol. 30, Princeton University Press, Princeton, N.J., 1970.
  19. S.N. Vasil’eva, Yu.S. Kan, Approximation of probabilistic constraints in stochastic programming problems using a probability measure kernel, Avtomat. i Telemekh. 80 (2019), no. 11, 93-107.
  20. M. Yousefi, K. van Heusden, I.M. Mitchell, G.A. Dumont, Model-invariant viability kernel approximation, Systems Control Lett. 127 (2019), 13-18.
  • Daniel Alpay (corresponding author)
  • ORCID iD https://orcid.org/0000-0002-7612-3598
  • Schmid College of Science and Technology, Chapman University, One University Drive Orange, California 92866, USA
  • Communicated by Aurelian Gheondea.
  • Received: 2020-11-20.
  • Accepted: 2021-01-23.
  • Published online: 2021-04-19.
Opuscula Mathematica - cover

Cite this article as:
Daniel Alpay, Palle E.T. Jorgensen, New characterizations of reproducing kernel Hilbert spaces and applications to metric geometry, Opuscula Math. 41, no. 3 (2021), 283-300, https://doi.org/10.7494/OpMath.2021.41.3.283

Download this article's citation as:
a .bib file (BibTeX),
a .ris file (RefMan),
a .enw file (EndNote)
or export to RefWorks.

In accordance with EU legislation we advise you this website uses cookies to allow us to see how the site is used. All data is anonymized.
All recent versions of popular browsers give users a level of control over cookies. Users can set their browsers to accept or reject all, or certain, cookies.