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

Alpay, Daniel; Jorgensen, Palle E.T.

doi:10.7494/OpMath.2021.41.3.283

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Tytuł artykułu

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

Autorzy

Alpay Daniel , Jorgensen Palle E.T.

Treść / Zawartość

Pełne teksty: $C:\Users\Danuta Ryś\Documents\OpMath.2021.41.3.283.pdf [zdalny]$

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DOI

10.7494/OpMath.2021.41.3.283

Warianty tytułu

Języki publikacji

Abstrakty

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.

Słowa kluczowe

reproducing kernel positive definite functions approximation algorithms measures stochastic processes

Wydawca

AGH University of Science and Technology Press

Czasopismo

Opuscula Mathematica

Rocznik

2021

Tom

Vol. 41, no. 3

Strony

283--300

Opis fizyczny

Bibliogr. 20 poz.

Twórcy

autor

Alpay Daniel

alpay@chapman.edu

Schmid College of Science and Technology Chapman University One University Drive Orange, California 92866, USA

autor

Jorgensen Palle E.T.

palle-jorgensen@uiowa.edu

The University of Iowa Department of Mathematics 14C McLean Hall, Iowa City, IA 52246, USA

Bibliografia

[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 Mathe-matics/Ouvrages de Mathematiques 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 fonctionnel le, 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 Mathematiques, 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, Classics in Mathematics, Springer--Verlag, Berlin, 1995, Reprint of the 1980 edition.
[13] H. Meschkovski, Hilbertsche Raume mit Kernfunktion, Springer-Verlag, 1962.
[14] J. Neveu, Processus aleatoires gaussiens, Number 34 in Seminaires de mathematiques superieures. Les presses de l’universite de Montreal, 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. Scholkopf, 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.
[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 Mathe-matics/Ouvrages de Mathematiques 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 fonctionnel le, 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 Mathematiques, 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, Classics in Mathematics, Springer--Verlag, Berlin, 1995, Reprint of the 1980 edition.
[13] H. Meschkovski, Hilbertsche Raume mit Kernfunktion, Springer-Verlag, 1962.
[14] J. Neveu, Processus aleatoires gaussiens, Number 34 in Seminaires de mathematiques superieures. Les presses de l’universite de Montreal, 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. Scholkopf, 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.
[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 Mathe-matics/Ouvrages de Mathematiques 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 fonctionnel le, 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 Mathematiques, 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, Classics in Mathematics, Springer--Verlag, Berlin, 1995, Reprint of the 1980 edition.
[13] H. Meschkovski, Hilbertsche Raume mit Kernfunktion, Springer-Verlag, 1962.
[14] J. Neveu, Processus aleatoires gaussiens, Number 34 in Seminaires de mathematiques superieures. Les presses de l’universite de Montreal, 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. Scholkopf, 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.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2021).

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Bibliografia

Identyfikator YADDA

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